A gîte is just the French name for an independent holiday home available for rent. They are generally old farmworkers' cottages or converted outbuildings and barns full of French character, and not in a bad way! They are great if you are in a larger group and can work out incredibly cheapl. Here's a few tried and tested to get you started: www.maisonbleau.com, www.thehouse.fr, www.gite.forges.free.fr and Le Clos Du Tertre.
Gîtes de France is also a good place to start looking.
The explosion in the popularity of Airbnb in the last couple of years means it's now a really good way of finding not just good but interesting properties to rent in Fontainbleau. The majority of places come with reviews so you'll know exactly what you're getting. It's a fantastic website too. Definitely worth a look.
A new hostel has recently opened in La Chappelle-la-Reine, Fontainebl'hostel. It offers dormitory and private rooms as well as the option to camp in their garden with full use of the kitchen and common room. There's also the option to rent boulder mats too and it's right next to a Patisserie which is a definite bonus!
There are lots of campsites scattered around the forest. Two popular campsites are La Musardiere at Milly la Forest and Le Pres at Grez sur Loing.
La Musardiere has a swimming pool and is within walking distance of bouldering so a good choice if you don't have a car. It is a bit more expensive that some of the other campsites in the forest.
Le Pres at Grez sur Loing is a large campsite with an english speaking owner. You have the option of various plots or ask to pitch up in their field area which means you can have fires too. A little shop on site has the basics and a BBQ area you can use.
There also a campsite at Malesherbes, www.camping-iledeboulancourt.com. Open all year and the owner speaks good English. Facilities include an indoor kitchen/lounge.
You'll find a lot more campsites on www.campingdefrance.com