Ask around if you do not have access to a computer. If you know the song title, you may find people around you who know the artist's name. Asking can't hurt.
Log onto your computer and launch iTunes (if installed on the computer). Click on "iTunes Store." In the search bar, located in the upper right-hand corner, type in the song name. You are going to be presented with all of the songs with the same name currently present on iTunes. Look for an artist name that looks familiar. If none of them seem familiar click on the "Play" button next to the song titles. This plays a 30-second sample clip. Try out the songs until you find the song you are thinking of.
Type in the name of the song into any search engine such as Google, Yahoo or Bing and click "Search." A list of articles relating to the song name is going to appear. Inside the articles the artist's name is most likely going to be stated.
Head over to either RollingStone.com or Billboard.com. These are two different sites that have a large search feature for music. Type in the song title into the search window and click "Search." You are going to be shown all of the songs to ever have the same title in addition to the artist names.