The CD-ROM format is the format used for store-bought, pressed glass CDs. It's the format CD players are designed to read and therefore ensures maximum compatibility across CD-reading devices. However, it isn't possible to produce a CD-ROM format disc at home because the manufacturing process requires a large scale press. To use this format for your music, contact a professional CD manufacturer.
CD-R is as close as you can get to a home-production equivalent of CD-ROM. Instead of storing data in pits and holes in tracks, like a CD-ROM does, CD-R format discs are produced by forming bubbles in the disc tracks using a laser. Once a CD-R is burned, it can't be overwritten but its close resemblance to CD-ROM means most CD players will be able to read any music written to this format.
CD-RW is a relatively modern CD format that allows a disc to be recorded to, wiped clean and then recorded to again. Although music burned to a CD-RW should in theory be readable in other drives, the structure of a CD-RW disc is sufficiently different to cause problems in some CD players, including some modern car stereos. It's therefore advisable to use CD-R or CD-ROM format discs if possible to avoid compatibility problems.
Audio CD-Rs, sometimes known as "music blanks," are blank CD-R format discs that are supposedly designed for recording music. However, note that there's no significant difference between an audio CD-R and a standard CD-R and you can use both to burn music to a CD.
Not every CD player is able to read a home-burned CD. In most situations this is due to the home-burned CD having a scratched or otherwise damaged surface such as the side with the music burned to it having fingerprints on it. In most situations, cleaning the CD will allow it to work, but wherever this doesn't work, the CD player may simply be incompatible, in which case use the CD-ROM format.