Another option, which is what I have been using, is Audacity, which records and plays, and also has a lot of more-advanced editing functions that I may never take advantage of. Audacity is free open-source software, and is available
here.
The only downside to Audacity is that it will not directly export MP3 files. Instead it exports in the OGG format.
Ralesk has also indicated that Goldwave is another good option, but I haven't yet tried that software.