How do I know whether to use 'qui' or 'que'?

Qui and que are relative pronouns, which means they refer back to a noun (a thing or person) you've just been talking about, in order to add some information about it. In English, we use which or who/whom. 

eg. I was wearing a scarf, which was blue. (which refers back to the scarf) 

A common misconception is that qui = who/whom and que = which, but this is not always true. 

The important thing to consider is whether your noun is acting as a subject or an object. 

If it's a subject, it will be 'doing' the verb
eg. I was stroking the dog, which was barking. (The dog was barking) 
      I am talking to Lisa, who likes ice cream. (Lisa likes ice cream)
      We saw Joe, who is very tall. (Joe is very tall)

In English, we use who or which, depending on whether it’s a person or an object. In French, we use qui. Simple as that.

If it's an object, it will be having the verb done to it. 
eg. He said hello to the dog, which I was stroking.  (The dog was being stroked)
      I stole the ice cream, which she was eating. (The ice cream was being eaten)
      He waved at Joe, who(m) we were looking at. (Joe was being looked at)

In English we use whom (though nowadays a lot of people just say who) or which, depending on whether it’s a person or an object. In French, we use que. Simple.

SO, forget the relative pronoun for a second and try putting your noun with the extra info you want to add about it. For our first example, this means taking the 'scarf', and putting it with 'was blue'. This gives us ‘the scarf was blue’.

The scarf is clearly being blue all on its own, with no one else involved, so it's doing the verb and is therefore the subject. 

So in French, our original example becomes:

Je portais une écharpe, qui était bleue.

More examples:

I visited the Eiffel Tower, which is very big.
J’ai visité la Tour Eiffel, qui est très grande.

I was talking to Lisa, a girl who(m) I met yesterday.
Je parlais avec Lisa, une fille que j’ai rencontrée hier.

He served some pasta, which he had made himself.
Il a servi des pâtes, qu’il* avait cuisiné lui-même.
*(NB If que is followed by a vowel, it runs in and becomes qu’. Qui never does this)

We decided to talk to Joe, who has a lot of good ideas.
Nous avons décidé de parler avec Joe, qui a beaucoup de bonnes idées.