Diastereoisomers are stereoisomers which don't have an enantiomeric relationship. For example diastereoisomeric relationship have these two structure of 2-chloro-3-hydroksy-2,3-dimethylpentanoic acid:
These molecules are of course stereoisomer but they aren't enantiomers because they don't have mirror-image relationship (enantiomer of first molecule would have S,R configuration on stereogenic centres, enantiomer of second molecule would have S,S configuration on stereogenic centres.
Diastereoisomerism can also occur in molecules which don't have chiral carbon atoms at all. For example - these compounds:
are also diastereoisomers.
meso-compounds
Well, meso-compound is compound which has chiral carbon atoms but it itsef isn't chiral. For example:
are meso-compounds. These two structures aren't enantiomers because it's possible to put first structure on second. You can easily recognise meso-compound because it has axis of symmetry:
racemates
Well, racemate is mixture of enantiomers of one compound in which there is 50% of first enantiomer an 50% of second enantiomer. Racemate has no optical activity.