Risa Mizuno is a girl Takaki met in Tokyo. They met a couple of times due to their work and started going out after a while.

In the movie, Risa and Takaki were together for awhile and Risa started sending text messages to him, inviting him to hang out, but Takaki did not reply back since he still has feelings for Akari.

When Akari moved on and got married, Takaki tried to move on and started dating Risa Mizuno, but eventually broke up.
Risa's background story is explored in more depth in the manga and light novel. Apparently she and Takaki were dating for more than four years. They went on several trips and dates together; but besides from her attempts to move forward with their relationship, they stayed the same. This becomes explicit when she invites Takaki to meet her parents and he turns her down. He would use an excuses, but she thinks he was avoiding her because she tried to rush their relationship. Later on, Takaki admits that he had commitment issues due to his past since he really believed someday he would be together with Akari again, so he couldn't truly love anybody else. Afterwards, Risa decides to break up with him and move on with her life.

In chapter 9, after breaking up with him, she says, "I hope we meet again, in some new way, as new people, and that something new starts."
(Source: xbobx, AkatsukiUlquiora)