Okay, I'm only going to go with those who have appeared in the new series, because otherwise there are far too many.
Donna Noble - I loved her personality and her chemistry with the Doctor, and I loved that they were explicitly just friends. After Rose and Martha, we needed someone to just be his pal. And you can see David Tennant's and Catherine Tate's real-life friendship shine through. They have great, hilarious chemistry wherever they appear together in British entertainment (and they do appear together a lot), and Doctor Who is no exeption.
Rory Williams - Just awesome. His loyalty, determination, and unquenchable love for his girl are the most amazing things. As a companion, he's hilarious, and as a character, he's impressive, yet still manages to maintain his every-man status. He never ceases to make you feel like you're watching an average guy, but his qualities are so far beyond average. What girl wouldn't want to end up with a guy like Rory?
Rose Tyler - I can't help it. She doesn't have any particularly outstanding qualities as a character. There's nothing altogether special about her. And maybe that's the best part. She, like Rory, is just the average girl you'd meet on the street. I think her strength is in how well written the character is, and how well the chemistry worked between Billie Piper and her costars, both Eccleston and Tennant. They made the romance so sweeping and epic that even though she would be actually a really boring character on her own, I couldn't help but love her.
Amy Pond - I think this character is carried pretty much entirely on Karen Gillan's screen presence, because when I think of some of her lines of dialogue in writing I can't help but think that they would be obnoxious. But Gillan carries Amy and gives her a lot of life, and I love Amy.
River Song - Okay. I really hated this character at first. I couldn't stand her, and I felt like she was being foisted onto us as this ridiculously smug love interest. But they've gone interesting places with the character, and I've come to really enjoy her whenever she shows up.
Also, I didn't vote for him, but I think he deserves an honorable mention: Wilfred Mott - Since when do old guys get to be companions? Since WILFRED EFFING MOTT, that's when. I love how subtly they introduced this character, and how kind and wise he is while at the same time giving you the impression that he's out of his league when it comes to tagging along with the doctor, and he knows it. He's such an interesting character to me. There are aspects of his character that are strangely childlike. He feels like a symbolic representation of the eternal child in all of us; no matter how old we are, there is still room for wonder, and he shows us that so well.