We know what affixes are. They come in two flavours: prefixes, which precede their stem, and suffixes, which follow their stem. Similarly, clitics are of two types: proclitics, which precede their host (not stems!), and enclitics, which follow their host.
Some definitions of clitics are based on apostrophes: a clitic is allegedly something added to an adjacent word with an apostrophe. This is clearly not a valid definition, since it makes reference to spelling, which is not part of language.
According to another definition, a clitic is not sensitive to what it is added to, as opposed to an affix, which is. An affix, a suffix like -ness can be added only to adjectives: happi-ness, mad-ness, not to verbs or nouns. The plural affix -s can be added only to nouns: cat-s, dog-s, not to adjectives or verbs.
A clitic is not so picky. The enclitic -’s (the full form of which is is or has) may follow any word category. Look at the following examples:
The boy's crazy.
The boy's gone home.
The boy I know's crazy.
The boy I know's gone home.
The boy who's lazy's also crazy.
The boy who's lazy's gone home.
The clitic follows a noun in the first two, a verb in the second two, and an adjective in the last two sentences. An affix cannot be so versatile.
In a clitic, the apostrophe marks the omission of a vowel: boy is/boy’s, could have/could’ve, she will/she’ll. The vowel of a clitic is not necessarily omitted. Look at the following sentence:
John can do it
In it can and it are clitics, but nothing shows this in the spelling. They are clitics because they are unstressed: [dʒónkən dúwit]. The two words John and can are pronounced as if they formed one word, like, say, sunken [sə́ŋkən], and similarly do and it are pronounced as one word, which rhymes with the name Hewitt [hjúwit].
So a clitic is a word, in fact, a function or grammatical word, that loses its stress and with that its status of a word from a phonological point of view. However, syntactically clitics remain words. Words can be inserted between a host and a clitic:
John <also> can do <all of> it
This is not something that can be done in case of a stem and an affix.
The same holds of proclitics: the articles the or a/an are proclitics in most of their occurrences, they are unstressed. A boy [əbój] is pronounced like the word annoy [ənój]. And words can be inserted between this proclitic and its host: a lazy boy.
Interestingly, what we called the genitive “suffix” behaves somewhat like a clitic. It is not picky about what it attaches to:
the girl's hat
the girl I saw's hat
the girl I saw yesterday's hat
the queen of England's crown
(And, it is spelled with an apostrophe ☺)