Dumbbells
Free weights are incredibly versatile and very functional when it comes to working out at home. You can use dumbbells for thousands of different upper and lower body exercises. To get started I would suggest you purchase 2-3 sets of different weight sizes. Light, medium and heavy- based on your fitness level. You can't go wrong with investing in a few sets of dumbbells as they are fairly inexpensive as far as exercise equipment goes and take an extremely long time to wear out, especially in a home setting.
Stability Ball
The stability ball is another inexpensive piece of equipment that goes a long way. From core exercises to simply using it as a dynamic seat in front of the TV, the exercise ball can be very useful. Balls come in different sizes however, so make sure the size fits your body comfortably.
Resistance Bands
Resistance bands are a nice alternative to weights and can be easier on joints than dumbbells. I prefer the bands with handles because they can be a little more comfortable for the hands and wrists.
Your Own Body
Don't forget about the exercises that can be done simply with your own body weight. For most core exercises as well as pushups, dips, squats, lunges, leg raises, etc. all you need is proper form and your own body. Also, don't forget about all of the cardio that can be done without any fancy equipment--you don't need a treadmill to simply walk, run, and climb hills.
While not absolutely necessary for a great workout, the next couple pieces of equipment make working out a ton of fun...at least for me anyway :)
This simple strap mechanism uses your entire body, especially the core, to stabilize while still targeting a specific muscle group, which teaches your body to work as a whole (like you would in daily activities) rather than each muscle in isolation. Changes in body placement increases or lowers the resistance of the exercise. Keep your core tight on the TRX and no matter what you do your core will feel it too afterwards. I absolutely love my TRX and it always keeps workouts interesting.
Bosu Balls
The Bosu (the half ball) is another piece that forces you to use your stabilizing muscles. I use my Bosu mostly for core exercises when I want a fun challenge but it can also add a dynamic element to squats, lunges, pushups and more.
Kettlebells
These fitness tools have been around for ages and have more recently come back as popular functional training equipment. They can be used similarly as dumbbells (while being a little easier on the wrists during some lifts) or by controlled swings used for cardio and power training.
Regardless of what equipment you are using or what exercise you're doing, be sure you are using proper body mechanics and are able to perform the movement with correct form.
Good luck picking out the equipment that is right for you, and as always...have fun!