Mikado and Anri met as fellow classmates/class representatives and quickly developed a friendship because of it. They, along with Masaomi would frequrenlty hang out after school around the city.
In the light novels, after Masaomi leaves the city, Anri begins to call Mikado by his first name and the two of them begin to spend much more time together to the point where most in the school assume they have started dating. Indeed, they have even become somewhat famous around the school for it. However, they both deny these sentiments and insist on being only friends.
From the beginning, it is clear that Mikado has interest in being more than just friends with Anri and has made attempts to further this goal at varying points. However, his naturally timid nature often prevents him from expressing his emotions clearly. After Masaomi leaves, Mikado decides to avoid entering into an actual relationship with Anri because he wants everything to be as it was for Masaomi when he finally returns.
From Anri's side of the relationship, it's clear that Mikado's (admitedly somewhat feeble) advances towards her are starting to have an effect as she is shown blushing during her date with Mikado in episode 25 but, like Mikado, she doesn't want their relationship to advance too far but for reasons detailed in the section below.
In chapter 4 of Durarara!! SH's first novel, it is revealed that Mikado and Anri have officially began dating.
In the light novels, the reason Anri tries to avoid a romantic relationship with Mikado (or with anyone for that matter) is because of Saika. The way she explains this is that, Saika fills its host with an uncontrollable feeling of love that manifests as a desire to cut people (since, as a sword, that's all Saika knows how to express anything). Since Anri locked away all her emotions after her parent's died (specifically love), Saika merely balances out the emotions she has locked away which is how she is able to keep the spirit of the sword from going berserk. However, due to her time spent with Mikado, her emotions are beginning to reawaken and as a result, gives Saika more power.
Bottom line: Anri is afraid that, if she allows herself to fall in love with Mikado, it will give Saika enough power to take full control and cause her to hurt him. This fear appears to be well founded as Anri seems to have more and more trouble keeping Saika in check as the novels progress.
As for Mikado, things get more strained after the Toromaru incident in volume 05. After seeing Anri save his life not once but twice with Saika, Mikado begins to realize how useless he is towards his friends, and resolves to become stronger. This causes Mikado's darker half to emerge and he begins acting coldly towards many of the people he knew, Anri included. When Anri confronts Mikado about this, he coldly brushes her off and even breaks their promise about keeping their secrets until Masaomi returned. He points out how she kept Saika a secret (he saw her use it during her fight with Varona) and tells her to stay out of Dollars affairs.
When Anri tells Masaomi about this, the two of them resolve to bring Mikado back to his senses.