These projects have badges. Visit their pages for the explanations because they aren't all the same.
Asteroids@home
Bitcoin Utopia
Citizen Science Grid
Collatz Conjecture
Enigma@home
GPUGrid
Milkyway@home
NFS@home
NumberFields@home
OProject@Home
PrimeGrid
Radioactive@home
theSkyNet POGS - the PS1 Optical Galaxy Survey
World Community Grid
WUProp@Home
Yoyo@home