First, check out Marriott Benefits. It has a nice chart showing all the benefits of each level, including right under the "overview" the number of nights needed Silver, Gold and Platinum. (Others have already chimed in with the info, but if you want to see it on Marriott's website, the above link should help.)

Second, in the upper left-hand corner, you should see a menu with a "My account" link. Click on that and you should be taken to a nice page that shows how many nights you have for this year and even how many you need to reach the next elite level. (Note that if you aren't logged into MR yet, you will be prompted to login before you see your info.)