There's lots of reasons. For starters, some pop stars release different singles in different countries, which can end up with them having different "careers" in different places.
Sometimes there's a certain "event" that happens which boosts an artist's popularity hugely in a country. For example, maybe an artist collaborated with another artist that's big there, or maybe they're married to a celebrity from that country or something (e.g. Katy Perry was in the news a lot in the UK when she was in a relationship with Russell Brand).
Or maybe they do songs with influences from that country's music scene. For example, country music is big in the US, and Taylor has that genre's influence in many of her songs.
Rihanna has UK music's influence in many of her songs, for example dubstep (moreso than most other pop stars). And Rihanna has collaborated with the likes of Chase & Status, Calvin Harris, Emeli Sande, Naughty Boy, Labrinth, Alex Da Kid, and also appears on X Factor a lot, which is what makes her very popular here. The key moment where it happened was just after Loud was released and she was all over the news because of the Ofcom investigation into her (and Christina Aguilera), and she ended up with 3 songs in the top 10 at the same time and her album sold more here than it did in the US. That was the exact time I'd pin-point it to when the UK became her biggest country.
The UK has also become Nicole Scherzinger and will.i.am's main markets because of reality TV shows and also their relationships/friendships with UK celebrities (Lewis Hamilton and Cheryl Cole).
However, I went to the Caribbean on holiday once, and I'd say Rihanna is just as popular in the Caribbean as she is here, maybe even moreso. People like Shakira were very popular there too.
Other times it can be more simple, maybe a certain country happens to have lots of major radio stations willing to playlist said artist, which makes their songs sell better in that country, which in turn makes radio even more willing to play them (and it spirals out). Sometimes an artist has a single that flops in most countries, so they lose momentum, but there's one particular country where that single manages to be a hit, so they can keep their momentum going in that country.
Also, some artists have different record labels managing them for different countries, so one of the labels might promote the artist better in that country.