Jasper's link should spell it out.
Siver Elite....10 nights earned in a calendar year
Gold Elite...50 nights.
Platinum.. 75 nights.
You earn nights by your overnight stays. You can also get a Marriott Chase Credit Card. They award nights, up to 15 per year, and 1 for every $3000 spent.
From time to time, Marriott may offer bonus programs for points and nights. Currently, there is a birthday night credit.
Lastly from me. If you have status with another chain or you intend to stay frequently over the next few months...you can call customer service and the may offer an accelerated program.
I believe there also is some type of status given for Marriott Vacation Club Owners. Others on here can explain that better.
The easiest way to obtain silver elite, is to signup for one of the credit cards. I'd highly recommend getting the Marriott Rewards Premier card, as you get 15 elite nights per year, plus an additional elite qualifying night for every $3,000 you spend on the card. Beyond staying and matches noted above, you can also call MR and do a challenge. Challenges require you to stay X amount of times within 3 months, though. Another way to achieve elite status would be to host events at an Marriott property. Events may be as simple as a 1 hour meeting, to a full blown platinum wedding. Either way, you get 10 elite qualifying nights (not to mention 3 pts/$ spent) per event.
Hope this helps!
Aside from what the others have correctly already mentioned with the Marriott Rewards Premier card, you also get one free Category 5 Marriott Stay every year on the Anniversary of your card. That more than pay for the annual cost of the card. That night is credited to your account and is good for 6 months. I've also had Customer Service extend it for another 6 months when I've called and told them I hadn't been able to use it.
I feel bad for you for not getting that benefit , and only mention it to remind the people in the US of an advantage of getting the card here. I'll tell you what though, I'd gladly trade that 1 night stay that we get in the US, for the CL's being open outside the US on the weekends, that we in the US don't get I guess we all get some things we'd like to see EVERYONE get!