Multi-culturalism – “Quilt” - different groups live in the same society with completely different ethnic identities. Example - Canada, which was traditionally a union of English and French North Americans under a weak civic identity, and more recently is a union of indigenous and Asian ethnic groups in addition to ethno-Europeans.
Integration – “Mosaic” - different groups live side-by-side with a shared identity/culture. Over time, they mix together, and the shared identity/culture changes to reflect this. Example - Some metropolitan areas of the United States today, with vastly different ethnic groups but all sharing a cohesive ethno-civic identity as ‘Americans’.
Assimilation – “Melting Pot” - different groups live side-by-side with the same (or closely-similar) identity. Over time, minority groups are absorbed into the host culture or a constructed civic identity. Example - the United States up until the 1960s, which was a melting pot of different north-western European ethnies.