This is a very extensive list of course...personally I see people who travel with the Doctor regularly (i.e. Rose, Martha, Donna, Jack, Amy, Rory, Clara and River) as 'companions' whilst all the others would be 'assistants'.
My favourite companion is still Donna. She was superbly acted and diverse..could switch from comedy to raw emotional drama in one scene to the next. Her exit was tragic, but in its own way that made it memorable. My only problem was the false sense of significance about her character ahead of The End of Time which turned out to be a red herring - I liked that she got a slightly happier ending than before that didn't contradict, but it was such a tease.
Clara has quickly become my second favourite companion, pushing Rose into third and Martha into fourth. Clara just has an energy and optimism that I usually don't like in a character as it makes them quite shallow, but Jenna Coleman seems so naturally like that that I just love it. It meant that come scenes that pack a more emotional punch, it's much more unsettling and out of a comfort zone that makes for real drama.
In terms of the 'assistants' I have soft spots for both
Jackie and Wilf in particular. Adelaide Brooke is also fantastic as a one-off but don't think she would have worked as any more than that.

River is kind of her own thing, a bit more companion than assistant I'd say, but I love her too. Let down a bit by a disappointing plotline in Series 6 and just a few too many catchphrases that turned both her and Eleven into walking caricatures, but I love her character all the same and it's always a treat seeing her back.