Any solution used to clean vinyl Lp's is commonly known as record cleaner. The main stipulations for an effective record cleaner is that is is liquid, and able to wet the vinyl, not bead up on it. The solution must also be somewhat solvent, so as to remove all the dust, dirt, oil, and other gunk deposited on the record surface and in the grooves. The liquid solution is applied to the record, allowed to set for a few seconds, and then wiped up with a soft brush, carbon or velvet being popular choices.
Debate has raged over the best recipes for record cleaner,
usually at an arcane, academic audiophile level. For the purposes of someone who just picked up a stack of vinyl at a garage sale however, and just wants to play their records, a couple simple recipes are adequate.
For really old dirty records, you should first wash the record in warm, soapy water, rinsing thoroughly afterwards, and leaving to dry. Remember that this will kill the paper label if you are not careful, but if it's old vinyl which is on the edge of death anyway, it may not matter much to you. For regular day to day cleaning, a dependable solution is:
35% distilled water, 35% isopropyl or grain alcohol, 15% dish detergent, and 15% photoflo
You can also buy solution commercially, if you're really into your vinyl, both Gruv-Glide and Disk Doctor being excellent choices. Both come with a brush or pad to apply and wipe the record with. For most however, the simple and cheap homebrew solution will be more than acceptable when combined with a velvet pad or cloth. Never use this or any alcohol containing solution on your old 78's, they are shellac coated and will just plain dissolve if you do.
All the best with your records, and like I said, plenty of super audiophiles would consider my method sacrilegious and destructive, so proceed at your own risk.