I couldn't whittle my favorite down to just one alien, so I bring you two-three...ish.
My absolute favorite alien race has always been the Tenctonese from Alien Nation. Alien Nation was this alien buddy cop show that mainly followed the Francisco family (shown above sans the two people on the left) while they tried to make their place in earth society. The series also dealt with issues of bigotry. It was done quite well in my opinion and I'm always sad that there are only 22 episodes in the series. Oh well, at least we have the movies.
My favorite Tencotonese specifically was Buck, the Francisco's son. He had this bad boy thing going on, which is always appealing, but it wasn't one dimensional. Even though he was a bit of a dick, you felt genuine sympathy for him. Not only was he a teenager who had to deal with bullying in school because he wasn't human, but he also had the added difficulty of trying to stay in touch with his own culture while basically being stranded on a planet whose culture and people he didn't really want any part of. And on top of that his dad's a cop bent on integrating and cohabiting with the humans that populate the earth.
I think what was really great about his character is that the series showed him learning the errors of his ways and thinking, slowly but surely. By the end he still had residual anger but he wasn't that much of a bad boy. He had grown and matured. The show gave him quite a bit of good character development and, I think anyway, did a good job of showing all sides to the story that is Buck and Buck's life.
Plus Sean Six who played Buck is yummy, so there's that.
From the same series I'd also have to pick Moodri, the Francisco's uncle. He was this awesome hippy type character who had a pet goat. He could walk in salt water without getting burned (salt was like acid to the Tenctonese). He was funny and wise. I really wish that he had more screen time in the series and that we were able to learn more about him and his life. Unfortunately, if I remember correctly, he died in the series not to long after he was introduced.
Pictured above, Uncle Moodri being awesome. No other context is needed.
Last, but not least, my favorite alien is Quark from Star Trek.
I love how Armin brought his character to life. You can just tell by watching the DS9 series that he took a lot of joy in portraying him. I always paid attention a bit more whenever him and Odo were in a room. The way those two bounded off of each other, it was just a lot of great chemistry between the two actors. He was enjoyable in all aspects even when he was being a bit of a coward. Plus, for an ugly guy he got a lot of prime tail and you just have to respect that.
Thank you for letting me blab your ear off and completely fangirl out about my love for the above aliens. Now hop on down the blogger's trial: