Conformity is a change in behaviour or belief as a result of real or imagined group pressure. The minority yields to the majority.
Compliance

Publicly conforming to the behaviour or views of others but privately maintaining one's own views.
Based on a desire to fit in.
The change is weak and temporary- you only comply when the group are around.
Explained by NSI
Example of compliance

Walking into a party where people are drinking/doing drugs. You don't want to but you do anyway to fit in and be accepted by the group.
Identification

Adjusting behaviour and views to the group because membership of the group is desirable.
Public and private acceptance when the group are around. (ONLY when the group is present)
Explained by conformity to social roles.
Example of identification

In the army, you adopt the life of a soldier- when you leave the army, new behaviour and opinion will be adopted. You only conform in the group setting.
Internalisation

This is the truest form of conformity.
Publicly and privately agreeing to the majority.
This is a total change of beliefs after being exposed to the opinion of the group.
The individual makes up their own mind.
Based around a need to be right.
Explained by ISI.
Example of internalisation

Going to university where everyone in your halls are vegan. So you become vegan but carry it on even when at home, out of the group setting. You completely change your beliefs.
Differences: Motivating conditions

If you want to fit in, you comply, whereas if you are looking for the most appropriate way to respond in a given situation you'll need to internalise group views to seem credible.
Differences: Agreeing

If you only publicly agree to the opinions of the group and privately disagree, you comply, whereas if you completely shift your beliefs and agree both publicly and privately, you internalise group opinions.
Differences: Duration

If the change in beliefs is only temporary, only in the group setting, then you comply, whereas if the change is permanent, you internalise group opinions.