This website has a good explanation of events. I don't think there are any spoilers in it, as it is entirely focused on Season 1 - however, having not watched much of the series myself, I've tagged all quotes from the website as spoilers, just in case.
So to quote a few things:
Why did she leave?
All was not, however, as it seemed and it took the murder of Lilly Kane to shatter many of the illusions of Veronica's life, not least, those pertaining to her own family. Her parent's marriage did not have the strength to survive the public humiliation and loss of status brought upon them in the wake of Keith's dismissal for pursuing Jake as the murderer of his daughter. Lianne fell apart; she drank heavily.
Why did she really leave?
But as she peeled away the layers in her search for the truth about the death of her best friend Lilly, Veronica discovered that she was still far from knowing the truth about her own mother. Or herself. She found that before Lianne left, she had been sent photographs of Veronica upon which gun sights had been drawn, that the man responsible was Kane Software's head of security, Clarence Wiedman, that Jake may be Veronica's father and that Lianne hadn't just left town, she was on the run, begging Veronica in a rare call not to try to track her down, a sentiment echoed by Keith, who unlike Veronica is prepared to move on with his life without Lianne.
Why was the mother brought back?
She found her mother, drunk and terrified in a lonely bar. Reluctantly, Lianne told her tale...Such information was dangerous for the Kanes and the false alibis they gave. Lianne was convinced that she could not be with Veronica because to do so would risk her daughter's life. Lianne's conviction was reinforced by Clarence Wiedman following Veronica to the bar...One thing was certain. Veronica Mars was no coward and would not be terrorized into losing her mother again. With all of her savings, her investment for her future, Veronica placed Lianne in a rehabilitation center.
Why did Veronica kick her out?
Veronica's dream to reunite her family came true when her mother came home, but once Veronica discovered that Lianne had started drinking again, she put aside her childish dream. Her college fund squandered on a worthless investment, Veronica ordered her mother to leave. Lianne did, with money earmarked to rebuild Veronica's future.
So to quickly answer your questions - she is brought home by Veronica who wants to give her a second chance. She is then sent to rehab. We don't see that Keith brings her home, but rather that Veronica comes home and finds them both together.
Therefore, what has happened is left up to the viewer. Maybe she lured him into picking her up from the rehab she was supposedly at, maybe she arrived home on her own and found him. Either way, she is home and that's all the really matters.
This then leaves Veronica in the next episode to discover that she was using her water bottle to store alcohol and was therefore still drinking. Outraged at such a betrayal of her trust (it cost all of Veronica's college savings) she ordered her to leave.
Remember, given Veronica's strong independence, determination and investigative abilities, it makes sense in the context of the series for her to come from a broken/dysfunctional family where even in her personal life her detective skills are put to use on a constant basis. Therefore, even though this might seem like a loose end, I think it works (and many other series have done similar things, e.g. Ryan Atwood's mother in The OC.
