Good question. So many native speakers get this wrong. Comprise means 'contains', 'made up of'. You use it to identify the constituent parts of a whole. The property comprises four main buildings. The course comprises six face to face lectures and regular tutorials conducted by video conferencing.
Compose refers to the elements which combine to make a whole. Equal parts rum, lime juice and syrup compose the drink. This can also be expressed as 'the drink is composed of equal parts rum, lime juice and syrup' . So you can say that a larger thing 'comprises', or 'is composed of' smaller things. So to summarise:
a big thing comprises (contains) several smaller things
A big thing is composed of (made of) several smaller things
What you can NEVER say (yet I see and hear this often), is 'comprised of'. Nothing is ever comprised of anything