The pronoun 'on' can be tricky in French because it doesn't have an exact equivalent in English, and it is highly flexible.

It looks a little like the pronoun 'one' in English, and it can be used in this way.

En dînant avec le président, on ne devrait pas se moucher avec sa manche.
(When dining with the president, one should not wipe one's nose with one's sleeve.)

After all, it does take a third person singular verb in just the same way.

However, this is perhaps the least common use of 'on', which in the right context can be used in place of any of the subject pronouns (although it will always take a third-person singular verb).

In place of 'nous'

Most commonly, it replaces 'nous' ('we'):

On va au cinéma ce soir ? (Shall we go to the movies this evening?)

is the same as saying

Nous allons au cinéma ce soir ? (Shall we go to the movies this evening?)

('On' is used a huge amount to mean 'nous' in spoken French. The pronoun 'nous' is in fact becoming less and less common in spoken French.)

The following conversation was once heard coming out of a soccer match in which the referee made a disputed decision:

" On a gagné ! "  
  " Non, on a gagné ! "
" Non, on a gagné ! "  
  " Non, on a gagné ! "
" Non, on a gagné ! "  
  " Non, on a gagné ! "

and so on, all the way to the nearest café. This doesn't make any sense if you're still thinking that 'on' is used like 'one' in English, as a polite, modest way of referring to oneself. Remember, soccer fans are rarely polite, and never modest (especially to the opposing team).

 

('We won!'  
  'No, we won!')

not,

'One has won!'  
  'I say, I don't think so, old chap,
one has won!')

Other examples of 'on' replacing 'nous':

Quand j'avais ton âge, on n'avait pas d'ordinateurs, on n'avait pas de télé, on n'avait même pas de chauffage central, mais on vivait heureux. (When I was your age, we didn't have computers, we didn't have TV, we didn't even have central heating, but we were happy.)

In place of 'ils'

'On' is also fairly commonly used to mean 'ils', especially in those vague statements that convey rumours and popular wisdom.

On propose d'interdire le chewing-gum dans les écoles.
(They're talking about banning chewing gum in schools.)

On dit que les Français sont moins polis que les Américains.
(They say that the French are less polite that Americans.)

On dit qu'il va pleuvoir demain.
(They say it's going to rain tomorrow.)

Although 'on' can be used in place of the other subject pronouns as well, it is only commonly used to replace 'nous' and 'ils'

On in place of other subject pronouns

'On' can replace 'je', especially in more polite language, especially by royalty:

On n'est pas amusé.
I am not amused.

'On' can replace 'tu', especially trying to be warm and fuzzy and inclusive (especially to kids):

Est-ce qu'on a assez mangé, ma petite choute ?
(Have you eaten enough, sweetie-pie?)

'On' can replace 'il' or 'elle', especially in a more elevated style of French:

Vous avez parlé à la directrice ? Et est-ce qu'on vous a donné une augmentation? (So you spoke to the manager? And did she give you a pay increase?)

'On' can replace 'vous', especially in polite, written language:

On est injuste, Monsieur. C'est la première fois que je suis en retard.
(You are being unfair, Sir. It's the first time I've been late.)

But remember that these are all very rare. 'On' most commonly replaces 'nous'.