SF4 is the original Street Fighter IV, which came out in arcades first in… 2008? And later on PS3 and 360 in 2009, I believe. It only has 25 characters, and every character only has one Ultra.
SSF4 is Super Street Fighter IV, which has 35 characters (added Adon, Cody, DeeJay, Dudley, Guy, Hakan, Ibuki, Juri, Makoto, and T. Hawk), and each character can choose one of two Ultras. It also included new stages and several balance changes and tweaks that makes each character just slightly different. For example, Sagat was the best character in the original SF4, but in SSF4 he was not as good. SSF4 was only ever released on console, it was never in arcade form.
SSF4 AE is Super Street Fighter IV Arcade Edition, which has 39 characters (added Yun, Yang, Oni, Evil Ryu). Gameplay-wise it is the same as SSF4, but with a few more balance changes. Arcade Edition was released in arcades (as the name says) first, in late 2010 but just recently got a console release.