The interview process doesn’t reinvent the wheel. Typically, Google uses a two-stage interview process. The first stage is a phone interview (or, more often, a Google Hangout interview). The second stage is an in-person interview. Only finalists will make it to the second stage of the interview process.
The interviews themselves can vary slightly depending on how technical the job at hand is expected to be. For more technical positions, the interview process may include a shared Google Doc where the applicant can write code or algorithms and show them to the interviewer in real time. This system allows the interviewer to ask technical questions that require a demonstration of skills.
In general, Google is looking for candidates who excel in four different categories: general cognitive ability, leadership, role-related knowledge, and "Googleyness." Googleyness is basically a synonym for company culture fit and integration.
In the past, Google has been famous for asking off-the-wall interview questions like, "How many piano tuners are there in the world?" These questions were used to assess creativity, reasoning, and problem-solving skills -- among other things. In recent years, Google has pulled back a bit on the odd questions, though they still crop up from time to time according to applicants.