SLI = Running 2 or more Nvidia graphics cards in a single PC to boost graphics performance.
CrossFire = Basically the same as SLI, but with AMD cards.
What are the differences?
SLI only works with identical GPUs (2x GTX 770 for example), but the manufacturer (Asus, MSI, EVGA) doesn't matter.
CF is a bit less selective; You can Crossfire the HD 7970 with the R9 280X or the HD 7950. Everything will work as long as the cards use the same architecture.
Remember, having 2 cards doesn't double the performance; Some games get a performance increase of 80% and others will not benefit from it at all.
The problems?
SLI tends to have worse performance scaling than CF, but it has less problems like microstuttering. If possible, always get the...