Tara is loved for many reasons but I think the key is as many have already said in here, that viewers can identify with her.
The fact is, as the series progresses, it became more and more fanciful. Pretty much everyone, while well-characterized, took on this larger-than-life quality to them.
Then here comes Tara in Season 4. A completely new character and while she's a witch, the magic part is so minor and is really forgotten (by fans and by the creators themselves it seems) in favor of showing her off as a person.
Tara wasn't going around fighting monsters or sitting around discussing about the next inevitable evil to strike Sunnydale. (at least initially) She was just this adorably shy friend of Willow. In many ways, I believe this is commented on or it's said to be intentional, she's much like Willow at the start of the series.
As Willow grew closer and felt safer in her little group, Tara found the same kind of security with first Willow and then the entire Scooby Gang. This helped her blossom out because she was totally a shrinking violet at first.
Therein lies the human element that attracts so many people. I know I personally can relate to feeling nervous and awkward around people for most of my life until I found the right group and that helped me "come out of my shell" as one of my high school teachers even commented on.
The relationship with Willow is of course another big reason to adore her I think. Discounting Season 6, the couple was pretty much the only one to escape the horrors of the sort of soap opera drama that plagued every other relationship in Buffy.
Don't get me wrong, I loved the relationships in Buffy and it's one reason I prefer it to Angel. But it was nice, after seeing so many depressing flops (Buffy/Angel etc..) and comic relief (Xander/Cordy.....), to have something that really looked beautiful and could very easily work out in the long term.
Not that I don't enjoy dynamic characterization. No relationship is flawless and the first hints of tension in Tough Love was a nice change of pace. (made all the more tragic by what happened) I just don't think it was handled all that well come Season 6. It seems a lot of people started calling Tara a hypocrite because of her actions in that season and since I flatly refuse to ever watch that tripe again, I can't say if they're right or not.
BtVS' was pretty much 50% drama/socialization/characterization and 50% action. Tara was made for the former and it was there she flourished. So I don't really mind she had almost no presence in the action part.
So yeah, that's how I feel about her. I shall conclude with a picture.
^ this is all that is *adorable, agreeable, alluring, amiable, angelic, appealing, attractive, bewitching, captivating, charming, cuddly, delightful, desirable, enchanting, engaging, fetching, friendly, genial, lovely, lovesome, pleasing, sweet, winning, winsome
(thesaurus for the win)