Notice the main parts of a song. Most songs contain a chorus played a few times during the song which is usually distinctive from other songs. Remember, or write down, a few words from the song exactly as you hear them. Next, go to Google and type the words in in the Google search bar with quotes around them. This will often (if you got the lyrics correct) bring up a bunch of pages with the lyrics of the song. Most of these pages will have the artist name as well.
Dig a little deeper if you discover that there was more than one version of a song. Many artists will cover older songs, and a lot of times their fans may never know the song was a cover. If you discover that the lyrics pages list an artist of a song, but that artist doesn't sound like the singer, keep looking through the lyrics pages. Sometimes they will list which year a version was released, and if the version you like is newer, it is often safe to go with the latest version. Also, visit Wikipedia, which has millions of songs listed by their respective artists. It will also often have information on remakes of a song, which can help in your quest to find the artist you are seeking.
Go to the Amazon website if you cannot figure out which artist sings the version of a song you like. Look on the page for the Amazon internal search engine and select "Music." Type in the title of the song. When the selections pop up, you can listen to small samples of the songs, which will usually let you know whether it is the version you want or not. You can also use P2P download programs such as Limewire or Bittorrent to listen to the different versions of a song and figure out which artist sings the version you like.
Visit a local music/book store chain. Most of the large ones (Barnes & Noble, Hastings) will let you sample their music selection by using headphones that are kept in the store. If you have already narrowed down the version of a song you like to a few artists, you can just look for that artist's CDs until you find the song, then listen to them in-store. Make sure you ask the store employees if you need help to find the CDs.