Lithium Polymer is a category of Lithium Ion batteries.
More correctly: "Lithium-ion polymer"
LiPo's use polymer electrodes as opposed to liquid electrodes typically found in "Li-ion" labelled batteries. Although, a "Li-ion" battery (in consumer electronics) isn't necessarily using old technology, as Zim said; in airsoft they're just slapping cheap 18650s together, and you don't want that.
LiFePO4 (Lithium iron phosphate) was the other common lithium battery for airsoft use, but I don't think they hold up to modern LiPos.
"Memory" doesn't occur on any lithium batteries, and it doesn't really occur on NiMH batteries, either.
NiMH tends to develop voltage depression, a slightly different effect.
NiCd's get memory, but avoiding it is as easy as just cycling the battery when charging.
If you've got a stock gun with no mosfet, get a 7.4
If you're got stock a gun with a mosfet, check to see if it needs an 11.1 or not. Some of them do need an 11.1 to work properly.
The reason people get swelling on their LiPos is due to mismatched batteries for use.
Most swelling isn't harmful anyway, and LiPos tend to give you plenty of warning before going up in smoke.
I've got a writeup on batteries here:
http://airsoftcanada.com/showthread.php?t=156571