Rhyme
A thing often assumed to be known in vain, rhyme is the perfect coincidence of all the sounds from the [last] stress of two words to the end of them. In Romance languages the statement has exactly the same meaning with or without the term 'last'; although, due to the existence in English of secondary stresses, it does make a difference here: e.g. 'circulate' carries a secondary stress on the 'a', whereas the primary one lies on the 'i', so that it could or could not be considered to rhyme with 'fate', according to whichever definition we are pleased to pick. classic English poetry has systematically embraced the easier way, and assumes they rhyme, as in this example from Shakespeare (from Sonnet XXXVIII):

Of course, by referring to a poem as 'rhyming', prosody, as well as the layman, means that the last words of two (or more) verses are connected by rhyme; the occurrence of rhymes in the middle of verses will be covered, maybe, in the chapter devoted to advanced poetic devices.

Rich rhyme
Rich rhyme is a backward extension of ordinary rhyme. In less abstruse terms, this means the two sequences of sounds in the words start being identical before the last stressed vowel. Of course a rich rhyme is also an ordinary rime, and equally of course the vice-versa is not necessarily true. Rich rhyming isn't popular at all in English; it is, and very much so, in decadent French poetry, where it is very much the rule rather than the exception, as is shown in this deliciously incomprehensible sonnet, 'El desdichado' (the wretched) by Nerval:

Mirror rhyme
An even more perverted form of rich rhyme, mirror rhyme is an extremely rare device, in which a word phonetically includes the other one entirely: for example there is mirror rhyme between 'cart' and 'art', 'clerk' and 'lark', 'flower' and 'our'. To the knowledge of the present writer, there are no major examples of mirror riming in classic English poetry, although the very beginning of Keats' Endymion seems to contain more mirror couplets than statistics would allow; it was used extensively in a piece by Raimon de Miravalh, 'Aissi·m te amors franc' (this way love keeps me fair), and other Provençal troubadours employed it from time to time, and never by mischance, as may be seen in this fragment of Bernart de Ventadorn's most famous song, 'Quan vei la lauzeta mover' (when I see the meadowlark move):

Identity rhyme
Identity rhyme is simply the phonetic identity of two words. They can have the same spelling as well, but it is not required: for example, 'bat' as an animal and 'bat' as a club is an acceptable couple, as is 'all' and 'awl' or 'talk' and 'torque' and so forth; in poetry, this sort of rhyme is often obtained by combination of two words, as in 'thinking' and 'th'inking', and Petrarca fairly often irks his readers with the 'l'aura' (th'air) 'Laura' (his mistress) match.
This device is often called 'rime riche' by Anglo-Saxon prosodists who, however, make quite a mess of its definition.
Of course, identity rhyme can be the most banal of devices, when the words concluding the two rhyming lines have the same sound, spelling and meaning; otherwise, it is an extremely refined, rare and difficult technique. Couples of words with the exact same sound but different meanings are called 'ambiguous rhymes'.
As identity rhyme conveys an idea of stillness bordering on obsession, its main use is in sextains, and examples of it are to be sought there; occasionally, though, examples of it are found in simple couplets: in Racine's Phèdre, Aricia, in love with Hyppolitus, who is in turn coveted by his own stepmother, blurts out:

Virtual rhyme
This is a false rhyme that exists only in English and is strictly connected to the nature of iambic verse: since an ideal reader would stress all even syllables, lines are assumed to rime when their last even syllables (in a pentameter, the tenth ones), and whatever comes after, are matched. This excerpt from Shakespeare's sonnet CV falls in such case, at least for a contemporary reader:

Assonance and consonance
We have stated in the beginning that rhyme is a relationship that connects two words through the perfect coincidence of each sound from the last stressed vowel on. When instead of all sounds only the consonants are the same, the relationship is called consonance, while if only the vowels are the same, it is called assonance; the terms are commonly used to denote identity of all consonantal (or vocalic, respectively) sounds in two words, wherever the accent is.
These devices are often used by inferior scribblers to substitute for rhymes; you shall not have examples of that, though, for two excellent reasons, the first being that they are little agreeable, the second that all those the Present Author can think of are still under copyright.
More decent poets use them, instead, for several purposes, such as linking different groups of rhymes, or in specifically devised forms. Assonance, for example, is the base of several Spanish forms, and is the ground on which Spanish symbolists have tailored their free verse; in most poems of this type the same assonance is used every other line, throughout the whole work (the effect is emphasised more by the fact feminine words are used for this). An example from Antonio Machado: