Indirect object pronouns replace nouns that 'receive the action of the verb' in some 'indirect' kind of way. Usually, this means they are preceded by the preposition 'à':

Mon frère aîné a donné ma bicyclette à sa copine.
(My older brother gave my bike to his girlfriend.)


Mon frère lui a donné mes gants, ma casque et ma culotte de cycliste aussi.
(My brother gave her my gloves, helmet and cycling shorts as well.)
(or: My brother gave my gloves, helmet and cycling shorts to her as well.)

Here are the indirect object pronouns:

Person
Singular
Plural
1st
me (m')
nous
2nd
te (t')
vous
3rd
lui
leur

Examples:

Marie téléphone à sa soeur régulièrement, en fait elle lui téléphone tous les soirs et elle lui parle de sa vie à la fac. Elle aime aussi téléphoner à ses amis le weekend et elle leur parle de sa semaine.

Things to remember:

1)
Like direct object pronouns, indirect object pronouns always go just before the verb of which they are the object. (More on placement)

2)
Since indirect object pronouns replace 'à + noun', they are often used with a group of verbs that frequently introduce a noun with 'à'.

For example, to say

          'I speak to my brother every week.'

you say

          'Je parle à mon frère chaque semaine.'

because 'parler' uses 'à' to introduce the person being spoken to.
When this person is a pronoun, you use 'lui':

         'I speak to him every week.'

         'Je lui parle chaque semaine.'

So 'parler' will often be used with indirect object pronouns.  
So will the following verbs:

donner donner quelque chose à quelqu'un to give something to somebody
dire dire à qqn to tell smb
demander demander qch à qqn to ask smb smg
permettre permettre à qqn (à faire qch) to allow smb (to do smg)
promettre promettre qch à qqn to promise smb smg
répondre répondre à qqn to reply to smb
téléphoner téléphoner à qqn to call smb, to telephone smb
obéir obéir à qqn to obey smb

As a further example, compare the following verbs:

'appeler' means to call someone (by phone). It takes a direct object:

          J'appelle mon petit ami chaque soir.
          (I call my boyfriend every evening.)

'téléphoner' means the same thing, to telephone someone. It takes an indirect object:

          Je téléphone à mon petit ami chaque soir.
          (I call my boyfriend every evening.)


If you want to replace 'mon petit ami' with a pronoun, with 'appeler' you will say:

          Je l'appelle.

With 'téléphoner' you will say:

          Je lui téléphone.

Things to be careful about:

1)

English will mislead you. In many ways, English is a lazy language. The example above shows how you can say 'her' or 'to her': both are acceptable, even though 'her' looks like a direct object pronoun. French is more strict than English: if the pronoun is a direct object, you have to use 'le', 'la' or 'les', if it's indirect, you have to use 'lui' or 'leur'.

But the biggest problem most people have is that they aren't really aware if a pronoun is direct or indirect, precisely because you don't need to know in English. When writing or speaking, students of French tend (much to their teacher's chagrin) to think in English and translate those thoughts into French as they write or speak. They want to say

I give her flowers every day.

and conclude 'The French for 'her' is 'la', I'll write Je la donne des fleurs chaque jour', which is wrong. They fail to see the 'to' that is hidden in the English 'her'.

To avoid this, think in French! or failing this, when you write your French sentence, try 'undoing' the pronoun (in other words replace it with the longer noun phrase that it refers to) and see if your instincts tell you to include an 'à'. You should come up with something like

Je donne des fleurs à ma mère chaque jour.
(or whoever it is you give flowers to).

If there is an 'à', use 'lui' or 'leur'. The 'à' is the difference between 'la' and 'lui'.

lui = à + la or le (when 'la' or 'le' is a pronoun)
leur = à + les (when 'les' is a pronoun)

2)
'lui' and 'leur' do not change according to gender. You cannot stick an 'e' on the end of either of these; nor, in this case, can you stick an 's' on 'leur'.

3)
'nous' and 'vous' are the same here as the equivalent subject pronouns, direct object pronouns, and reflexive pronouns. You have to keep an open mind when you come across these words in your reading.

4)
'me' and 'te' (and 'm'' and 't'') are the same as the equivalent direct object pronouns and reflexive pronouns. See the warning about reflexive verbs in direct object pronouns.