A full answer would require the equivalent of a few Spanish course classes, so I'll just clear the basics up.
What you found are combinations between verbs and pronouns. These are not different conjugations. The pronouns are the shorter, unstressed, non-empathic forms that are technically known as clitic. A pronoun is enclitic if it's attached to the end of the verb, and proclitic if it's attached before the verb. (Proclitic pronouns in Spanish are not written together with the verb, but they're attached to it nevertheless: they behave as if they were part of the same word.)
There are a number of rules dictating when a clitic pronoun can and cannot precede or follow a verb. These can get fairly complicated but here are some:
- Clitics never precede infinitives: it's conocerte, never *te conocer.
- Clitics may precede or follow conjugated verbs (except in the imperative), but in modern Spanish they usually come before: la miro is OK and mírola is OK too, but nobody says mírola nowadays.
- Clitics always follow verbs in the imperative: it's mírame and vete, never *me mira, *te ve (note: these sound exactly the same as other forms of the verb which are not imperatives).
- Clitics also follow the gerund forms: mirándolos, yéndome, conociéndonos.
- If there are two clitics (because the verb takes two objects), the first is always the indirect object: it's te lo diré or decírtelo, never *lo te diré or *decírlote.
As I said, these are not different forms of the verb (conjugations), in the same way as him in I see him is not part of the verb: you just need to learn the verbs by themselves and then learn how to properly place the pronouns.