Speculah:The Time Travel Theory
This theory appeared along with the World Reset Theory, since they are very similar. Depending on the turn of events, it could be proven valid or invalid as soon as episode 3 airs.
This is a variation of the World Reset Theory. We suppose here that the dream Madoka has at the beginning of episode 1 is the future (or "a possible future" in the sense of the Multiverse Theory). We also suppose that the world is the way it is because of Madoka (that is suggested by lines like "If I could become such a helpful person...That sounds wonderful. Or so I thought.", end of episode 2)
Someone (be it Madoka, Homura or a third party) notes that the only way to change this outcome is to travel back in time and do things so that Madoka doesn't become a Magical Girl (doesn't that remind you of The Terminator ?). From here on, several variations were discussed as follows.
Variations
Variation A
It is Homura that wished to go back in time and stop Madoka before harm is done. That could explain why she was after Kyubey in episode one, why she already knows everybody, and why she warns Madoka many times. However, the major flaw of this variation is that Homura already "is" a Magical Girl, so her wish has already been granted. This variation doesn't explain the dreams Madoka has either. However, it could be possible that it is the future Madoka who used her wish to send Homura back in time.
Variation B
It is Madoka that wished to go back in time, in the hope of making another choice this time and avoiding the fatal outcome. This variation explains the dream, and is consistent if Madoka decided to become a Magical Girl (and hence use her wish) at the end of the dream. However, it doesn't explain why Homura knows her and try to warn her.
Variation C
Like Star Trek: The Next Generation "Cause and Effect" or Haruhi's "Endless Eight", either Madoka or Homura could be experiencing a Causality Loop where they are repeating the same days/weeks/months over and over, remembering little to nothing at the start of the loop. For Homura, she may remember little details in each loop, enough to possibly try things like influencing Madoka and friends to not make the contract, or trying to kill Kyubey before he can contract anyone else. For Madoka, after watching all of her friends or others die or be turned into witches at the end, she could make her wish to reset everything back to how it was, which brings her back to the morning she woke up after the dream in Episode 1, thus starting the loop all over again, but she does not remember anything but Homura.
Variation D
The Cramer Theory: In the original timeline Homura knew Madoka, and was essentially in Madoka's place of being the only girl in the group to not make a contract by the end of episode 4. Like Madoka, she is mostly on the sidelines and support, and idolizes Madoka not unlike how Madoka idolized Mami. Come to the end of the original timeline, Madoka sacrifices herself, or becomes a witch or something. Kyubey still survives whatever happens, and Homura takes him up on the contract offer. She wishes to be able to rewind time, which she can do at any time, whenever she wants, thanks to not being specific. Homura rewinds all the way back to before the beginning, and begins to change everything to get to her ideal ending, or the closest thing to it. Its basically like a VN, where she makes a bunch of choices, gets a bad end, reloads the last save(rewinds time) and tries again. Over and over again. She starts to having to make compromises such as letting Mami die in order to avoid a bad end later down the road. This also explains her "perfectness" and the nature of her powers a bit more, as well as her actions. She is so perfect and smart because she has already redid everything so many times. When she stopped Kyoko from killing Sayaka, she didn't do it because she was so fast, but rather because she redid it a thousand times failing each time and "reloading the last save" until she got it right. Similarly, when she chased after Sayaka's Soul Gem, she probably chased and failed to grab it several times. She knows everything that happens not only because of her original timeline, but also because she has experienced so many variations on the current timeline. Madoka probably has made the decision to become a Magical Girl (and thus leading to an automatic bad end) countless times by now, and it only through Homura's resets and new decisions, interruptions, and talks with Madoka that Madoka has strayed from that path. She probably even changed events from Madoka's distant past to change the way she acts now, to make her less likely to choose the contract and such. In episode 8, when Homura breaks down on Madoka asking why she always has to sacrifice herself, she is actually talking about the numerous bad ends she has been preventing through her time rewinds.
Variation E
Similar to Variation B, Madoka makes a wish to go back in time, but it would Homura who actually goes back. This would explain why Kyubey mentioned that this was his "second time" that he got killed, since the first time was actually in a previous timeline. The problem with this theory is that it doesn't explain what happened to the Homura from the previous timeline, and how Homura became a Magical Girl in the first place.
The Reveal
Episode 10 shown that from the above, the most correct theory was Variation D. Homura wasn't a Magical Girl in the first timeline, but wished to save Madoka when she lost against Walpurgis Night. The rest is a repetition of timelines where Madoka either dies or becomes a witch, thus making Homura go back in time again. In one of them, Madoka, being a Magical Girl, personally asked Homura to stop her from becoming one.
Long story short, Homura has full-on Time Rewind powers, and has been using the extensively throughout the series. Because the show is not framed from her perspective, we only see the results and actions of her most recent choices at each decision. We also see it wear down on her, as she continues to become more emotional around Madoka and the subject of protecting her from the apparently "constant" sacrifices she is making. It also explains her power and abilities more efficiently than a simple "time freeze" would.