I am not an expert. This is what I've learned so far:
Smartweeds have flowers in a cluster at the end of the stem.
If the flowers are all single or in little clusters where the leaves attach to the stem you have a knotweed instead (Polygonum sp.).
Smartweeds have sheaths (ocreae) where the leaves meet the stem, and the upper edge of this sheath often has a fringe of hairs (cilia) growing up from it along the stem. These are very important in deciding which species you have.
Distinctive Species:
Jumpseed (P. virginiana)
Huge rounded leaves (the size and shape of a large lemon, though flat (obviously)). Marked with a dark triangle that is only on the center half of the leaf, not near either edge. This dark mark (chevron) fades by the time the plant flowers. Flowers are very spread out along a huge stem, at least a foot long. Flowers are white or green. Soon replaced by green fruit.
Asian Jumpseed (P. filiformis)
Same as Jumpseed but with dark marks extending to the edges of the leaf, still present when flowering, and flowers dark pink.
Halberd-leaved Tearthumb (P. arifolia)
Leaves large and triangular with the bottom corners very elongated and pointed away from the stem. The whole leaf about 4 inches long. The plant is covered in backward-curved prickles that can tear skin and cling to clothing. Flowers are only about a half a dozen, light or dark pink or white, in a loose cluster at the ends of long, bristly stems. This plant only grows in wet soil.
Arrow-leaved Tearthumb (P. sagittata)
Like the above, this plant is covered in backward-facing bristles that can tear skin and stick to clothing. The leaves, however, are basically long oval shaped, but with a triangular notch where the stem attaches to the leaf, leaving two very sharp, downward-facing lower lobes. The flowers are in roughly marble-shaped-and-sized clusters in white or pink. This plant only grows in wet soil and often forms very large colonies.
Mile-a-Minute: (P. perfoliata)
Leaves like a rounded equilateral triangle, with the stem attached just in from one side. Plant is a vine, climbing over shrubs and other vegetation thanks to backward-facing bristles that can tear skin and cling to clothing. Flowers are green in a tight cluster the size and shape of a grape, which turn into sky blue fruit, and seem to grow out of the center of a round leaf. This is an aggressive weed and does not need wet soil.
Nepal Smartweed (P. nepalensis)
Leaves like an elongated triangle but with the stem attached at the edge of the smallest side. The leaves are sessile (they don't have much of a stem). Plant is not bristly. flowers grow in a small, round cluster about the size and shape of a marble, out of the base of the uppermost leaf. Flowers are white. Leaves often have a red edge. Not found in NJ (that I know of) but present in several spots in upstate New York.
Most Common New Jersey species:
Low Smartweed (P. longiseta)
Dark pink, narrow, dense clusters of flowers max about 2 inches long, not nodding, with long hairs among the flowers. The clusters have the overall width of maybe a section of pipe cleaner. The lower part of the cluster may be somewhat interrupted. Sheaths have fringe that is nearly as long as the sheath itself. Grows in disturbed areas and does not need to be wet.
Also has smooth stems, smooth leaves, leaf stems under 1/4 inch, no hair on edges of leaves, largest leaves generally under 2 1/2 inches. Can have dark marks on leaves but these are not highly contrasting with surrounding leaf. Some hair on sheaths themselves but it is flat to the sheath, not sticking out. Usually does not form large patches of plants. (See also Lady's thumb, swamp, and dense-flowered smartweeds below).
Dotted Smartweed (P. punctata)
Interrupted, narrow clusters of white or green flowers, usually held upright or sometimes drooping, clusters usually not more than 3 inches long. Sheaths have fringe about half the length of the sheath.
Also has stems either smooth or with short hairs growing along the stem (not sticking straight out), Leaves can also be either smooth or with hairs pressed to the surface of the leaf. leaves have a stalk about 1/2 inch or less, and largest leaves are about 5 inches. Base of leaf is tapered to stem. If you look very closely at individual flowers you can see dots or bumps on them. (This is by no means the only species with these dots.) Must grow in wet soil. Often in large patches of several plants together. (see also waterpepper, swamp, dense-flowered, and Carey's smartweeds below)
Pinkweed (P. pensylvanica)
Broad, tight clusters of light pink flowers. Flowers a little larger than other smartweeds. Whole cluster generally not more than 1 1/2 inches long. Base of cluster never tapered. Sheaths never have any fringe. Upper stem is usually not smooth, but also does not have hairs sticking straight out from the stem.
Also can have white or darker pink flowers but this is rare. Leaves are large, the big ones are about 5 inches long and they have definite stems, about 1/2 inch long. Must grow in wet soil and often forms large colonies. (see also small and water smartweeds below)
Pale Smartweed (P. lapathifolia)
Very large smartweed with dense, drooping clusters of pale flowers. Clusters generally up to about 4 inches long. Sheaths never have any fringe. Stems noticeably broader than other smartweeds, commonly up to 1/2 inch wide at sheaths.
Also can have white or darker pink flowers. Leaves can be hairy near the margins or along the center vein. Larger leaves can be up to about 8 inches long and are often 6 inches. Leaves often have a dark triangular mark (chevron) in the center. Leaves have a definite but short stem, about 1/4 to 1/2 inch. Sheaths are large, up to about 1 inch long, and can have stripes.This plant does not have to grow in wet soil but usually does. It is annual and easy to pull out of the soil. (see also Carey's, stout and Far Eastern smartweeds and kiss-me-over-the-garden-gate below)
Other New Jersey Smartweeds
Lady's Thumb (P. maculosa)
Similar to low smartweed, but has tight, narrow clusters (not interrupted at the bottom of the cluster) of flowers that are both light and dark pink in the same cluster, with flowers that open wide, and without any hairs in the flower cluster. The sheaths have fringe that is much less than half as long as the sheaths themselves. The leaves usually have a dark triangular mark on them.
Also has longer leaf stems than low smartweed (over 1/4 inch), with larger leaves (over 2 1/2 inches for the largest). The leaves can have some hair, but it would be flattened to the leaf surface, or they can be smooth as in low smartweed. They are not linear but rather sort of long ovals as in most smartweeds. Does not grow in standing water, does not generally need wet soil, and does not usually grow in colonies.
Swamp Smartweed (P. hydropiperoides)
Similar to low smartweed, but with narrow, dense clusters of pale pink or white flowers always held upright (though they can be slightly bent). Clusters up to about 2 inches long. Grows in colonies in very wet soil, often in standing water.
Also has leaves fairly long and narrow for a smartweed, but not grasslike. Sheaths have a long fringe, nearly as long as the sheath, but there are no hairs among the flowers. Flower clusters appear tapered at the tip more often than in most other smartweeds. Flower clusters often interrupted toward base.
Dense-Flowered Smartweed (P. glabra)
Similar to low smartweed but with longer, very dense, narrow clusters of dark pink flowers, never interrupted, no long hairs among flowers, clusters about 2 inches long, or more. Leaves never with dark marks. Fringe on sheaths present but much shorter than sheath. Grows in wet soil. Uncommon.
Waterpepper (P. hydropiper)
Similar to dotted smartweed but with very long, drooping, narrow, highly interrupted clusters of often dark pink flowers (though it can be pale pink or white). Clusters are often more than 6 inches long. The lowermost flowers actually grow at the bases of the uppermost (very small) leaves and are surrounded by the sheaths. This is a fairly common and subtly pretty smartweed of wet areas.
Also even the largest leaves are generally under 3 inches, and the leaf bases are not as tapered as in most smartweeds. Stems and leaves never have any hair at all. Stems are often red (but not always) Like dotted smartweed, if you look very closely at the flowers you will see they are covered in dots or bumps. The sheaths have short fringe, well less than half as long as the sheath.
Carey's Smartweed (P. careyi)
A rather large smartweed with drooping clusters of flowers in white or light pink, but covered all over with extremely long hair that sticks straight out from the stem, unlike any other smartweed (though kiss-me-over-the-garden-gate has similar hair on sheathes only). Grows in wet areas. Uncommon
Small Smartweed (P. glabra)
Similar to pinkweed, but with very narrow leaves about three inches long but only 1/2 inch wide, with a pale midvein, rather like crocus leaves, or small blades of grass, but distributed along the stem as in other smartweeds. Flowers in a fairly dense, wide cluster but can be looser, narrower, and longer than pinkweed, and flower color can be darker or lighter than pinkweed. Sheaths have short fringe. Does not have to grow in wet soil. Uncommon.
Water Smartweed (P. amphibia)
Similar to pinkweed but growing in standing water. Leaves often floating on surface of water but can be freestanding as in typical smartweeds. Flower cluster very dense, up to 2 inches long, deep pink, a striking color, but lighter pink where the flowers are open. Flowers open in a ring around the cluster at a time and open flowers are much wider than buds, making the cluster look much broader in one section than the rest. Cluster does not droop though may be slightly bent. Stems are generally a little hairy but the hairs are pressed against the stem. Stems are often striped. Sheaths have no fringe. Leaves have short hair, pressed to the surface of the leaf, with long stems up to 2 1/2 inches. Leaves are large, 6 to 7 inches or so. Often forms large colonies in standing water. Must have wet soil. Uncommon but striking.
Stout Smartweed (P. robustior)
Similar to pale smartweed but sheathes have short fringe. A very large smartweed. Sheaths are often swollen at the base (but this is not the only species with swollen sheaths). Flowers generally white or pale pink in dense, long, drooping clusters. Leaves never have dark marks. This is perennial, so difficult to pull out of the ground, unlike the annual pale smartweed. The flowers are paler than in Far Eastern smartweed, but it is very dificult to separate these two with confidence. Does not need wet soil.
Far Eastern Smartweed (P. extremiorientalis)
Similar to pale and stout smartweeds. Sheaths have short fringe. A very large smartweed. Stems are often somewhat narrower than the other two large species. Flowers generally a mixture of dark and light pink in the same long, dense, drooping cluster. Leaves often have dark marks. Stems are hairy especially between leaves. Flower spikes often kinked. Very difficult to separate with confidence from stout smartweed. Does not need wet soil.
Kiss-Me-Over-the-Garden-Gate (P. orientalis)
A garden escape. A very large smartweed with dangling, drooping clusters of dark pink flowers. Clusters about two inches long. Sheaths are green (rather than papery and brown in most smartweeds) and where other smartweeds would have a fringe, they have a green collar, sticking out at a 90-degree angle from the stem. Sheaths are also very hairy, with hair that likewise sticks straight out from the stem. Does not need wet soil. Uncommon.