Solder paste is basically an alternative for regular flux-core wire solder, that's used for reflow soldering (put solder paste on pads, place components and heat). The solder is kept from creating bridges by surface tension (and flux), plus proper amount and reflow profile I assume. The major advantage of solder paste is that you can print it on the PCB using a stencil, beats tinning all those pads by hand (or machine). I'd guess you can get something similar by tinning the pads manually and adding liquid flux, but it's a lot more hassle.
My issue with solder paste is that it usually has a fairly short shelf life, which sucks for a hobbyist not doing lots of soldering. Many packages are designed for industrial dispensers, and some are hard to use without one.
Sparkfun has written a fair amount about their experiences with printing solder paste and reflow.