In addition to the I -IV - V blues progression and the ii - V - I Jazz progression, there's I - vi - IV - V, which I believe developed from the doo wop era in the late 50s. It was developed heavily by soul musicians during the 60s, who would use it in conjunction with the I - IV - V progression by using the Doo Wop progression for the verses and the blues progression for the choruses, or vice versa. Check out Sam Cooke's "Twistin' the Night Away" for a good example.
More recently, a lot of musicians relied on the I - V - vi - IV progression. For a period in the 90s it was difficult to find a song that didn't use this progression. Green Day's "When I Come Around," Semisonic's "Closing Time," and Blink 182's "Dammit" are all examples.
The Roman Numerals correspond to the scale degrees.
I - chord starts on the 1st scale degree.
ii - chord starts on the 2nd, and so on.
Most of the time, an upper case roman numeral, like I or V, indicates a major chord, while a lower case roman numeral, like ii or vi, indicates a minor chord. A roman numeral that is followed with a small circle or asterisk, like vii*, generally indicates a diminished chord.
Occasionally you'll see something like bII. The b means the chord is flat, so instead of playing the chord on the 2nd scale degree, you would start it on the note that is half a step below the 2nd scale degree.