How do you form the perfect tense in Spanish (say you HAVE DONE something)?

The most important verb when using the perfect tense in Spanish is 'haber'.

There is no immediate translation to English, but in this context it roughly means 'to have' (not to be confused with 'tener'!)

This is conjugated as follows:

I - he

You - has

He/she/it - ha

We - hemos

You (plural) - habéis

They - han

Next, we must add the action that has been done. In order to do this, we must take the relevant verb and find its past participle.
For example, if we wanted to say 'I have bought a coat', we would have to take the verb 'to buy' (comprar) and find the past participle ('bought').
For the majority of verbs there is a very simple trick:

-ar verbs

With -ar verbs, the trick is to remove the -ar ending and replace it with 'ado'.
'Comprar' thus becomes 'comprado', 'jugar' becomes 'jugado', and so on.

If we now bring everything together, we can work out how to form the above sentence:

He (I have) comprado (bought) un abrigo (a coat).

-er and -ir verbs

The trick is very similar for -er and -ir verbs. Let's use 'comer' as an example.
Similarly to -ar verbs, remove the -er ending, but this time replace it with 'ido'. 'Comer' thus becomes 'comido'.
-ir verbs also follow this pattern.

Examples

They have gone out - Han salido (salido = past participle of 'salir')

I have learnt - He aprendido (aprender)

We have spoken - Hemos hablado (hablar)

Exceptions

Unfortunately however, not all verbs follow this pattern. There are some exceptions which just have to be remembered. These are irregular.

Common irregulars:

Abrir (to open) - abierto
Decir (to say) - dicho
Escribir (to write) - escrito
Hacer (to do/make) - hecho
Ver (to see) - visto

Examples:

He has opened the window - Ha abierto la ventana

I have written - He escrito

They have done their homework - Han hecho sus deberes