Enantiomers are stereoisomers that are non superimposable with its mirror image. In general, those are compounds with carbon that has 4 different substituents but there are other cases: asymmetric sulfoxides, axially chiral compounds such as allenes or BINOLs or phosphorous compounds can be chiral as well. For each chiral compound, there is only one enantiomer.
Diastereomers are stereoisomers that are not enantiomers. In general, they have at least two chiral centers and different configuration at one or more.
To give you an example:
immagine you have compounds with two chiral centers with configuration RR. Then the enantiomer is going to have configuration SS and diastereomers either RS or SR.
If you look closer, you will see that the diastereomers RS and SR are in fact enantiomeric to each other.
Reactions can in general be called stereoselective or stereospecific.
a) stereospecific reaction is one that gives specific stereoisomers because of the mechanism of the reaction. The structure of the stereoisomers are determined solely by the structure of starting materials. Good example is halogenation of alkenes, due to the mechanism of the reaction, trans product is always obtained. Hydroxylation by OsO4 is another example, in this case, cis-diol is obtained. There is no chance you get trans-diol by the OsO4 oxidation. Diels-Alder reaction is also stereospecific. The typical SN2 reaction is also stereospecific, ther is full inversion of the configuration. (Of course, this is only on paper in real life the inversion might not be 100% but that just means there is competing mechanism other than SN2)
b) stereoselective reaction is one that could give you both stereoisomers in principle but only one is formed prefenretially (is "selected") due to the structural features present. For example, asymmetric reduction of ketones to chiral alcohols with CBS catalyst is good example. In principle, both enantiomers of the alcohol can be formed but structural features of the catalyst highly favors one enantiomer over the other.
And now it gets tricky: reaction can be both stereoselective AND stereospecific. Good example is asymmetric aldol reaction. In aldol reaction, you can get cis or trans product and each of them forms also enantiomer. The formation of cis vs trans product can be controlled by reaction conditions but it was found out that the structure of the enolate (E or Z) solely directs the formation of either cis or trans product. This is the stereospecific part - the structure of the starting material controlls the structure of the product. But the product is still formed as a racemate.
Now, if your enolate consists some chiral moiety, you not only get pure cis or trans product but also only one enantiomer will be formed because the chiral moiety that is present favors formation of one enantiomer over the other. There is huge deal of work on asymmetric aldol reactions in literature, I suggest reading a review or wiki on it because it nicely ilustrates this area.
To answer your other question: we determine the stereoiosmer composition by various methods. If your stereoisomers are enantiomers, you can use HPLC with chiral stationary phase to separate them and see if one is in excess. Usually, the method using racemic mixture is developed and then the actuall product of reaction is analyzed. If your stereoisomers are diastereomers, then common methods such as HPLC or NMR can distinguish between them since they are different compounds
Racemic mixture is always formed when there is no chirality present in the reaction. Be it chiral centesr in your starting materials, reagents or catalyst.
I hope I answered your questions, feel free to ask follow up questions