You can find the current playtime month in the Playtime, Plots & Events announcement. The playtime changes roughly every four weeks, unless delayed by a plot of some kind.

Magical Hogwarts is currently in the year 2001.

In order to advance in year on Magical Hogwarts all three schools use a system based on the number of posts you make in certain areas of the board (classes, offices, libraries). The study area and Quidditch boards also contribute to your post count, but spamming these two boards with threads to get posts will result in your threads being moved and posts being removed. When you make a post in any one of these areas, your post count goes up by one. After reaching a certain number of posts, your character ages. For a listing of the specific number of posts required to advanced for each year, please see the Classes/Graduating thread.

Because you need to have a certain number of posts to be a certain age, you may not have the same number of posts connected to your account as the number of posts you have actually made in character. If your character started out as a fifth year for example, you would have started with 600 posts as your post count.