Let's start with the station wagon, since it's been around the longest. It's quite simple actually. In most cases, station wagons (and hatchbacks) are just extendedversions of existing sedans. A wagon is going to feel almost exactly like a sedan from the inside. It will most likely share almost 100% of the chassis and drivetrain components with it's sedan sibling. The usefulness of a station wagon is obvious to someone who wants to drive a car, but with a little extra storage.
At the other end of the spectrum is the SUV. A sport utility vehicle is intended to be very versatile, offering lots of usage options. They are generally built on truck-like frames with rugged truck-like suspension components. This provides the SUV with lots of ground clearance, making it a good option for off-road use. You'll sacrifice some ride comfort if you go with an SUV. It's basically a closed-in pickup truck, with a few extra seats. If you like the utility and ruggedness of a truck, but need protection for your cargo, and lots of seating, an SUV is the ideal choice for you
Now let's talk a bit about the crossover. From the outside, a crossover generally looks like a an SU-, well actually more like a statio- hmm, I guess it's really more of a mini- ah ok I give up! The crossover really is designed to take advantage of features from several different categories of cars. The goal is to offer sedan-like comfort and handling characteristics with a little bit more utility and space than smaller hatchbacks and wagons. The chassis and suspension of the crossover will resemble that of a sedan. Ride height is somewhere in between a sedan and a true SUV. This gives you a bit more ground clearance, but still maintains decent handling and comfort. Inside, the crossover will feel like a wagon or small SUV. They are generally taller than a sedan, with cargo offerings somewhere between a wagon and a minivan. Compared to an SUV, a crossover will generally be lighter weight by the nature of it's construction, which can improve fuel economy by allowing the use of smaller engines.
In my opinion, the crossover will appeal to shoppers looking for something new and trendy. It's certainly got its benefits, but it's not going to excel at anything. It's in the middle of the car - SUV spectrum, so you're going to get pretty average performance in general. It's not intended to drive like a sports sedan, but you can't really beat it up off road. It's not generally going to be very compact, but you won't get the most cargo space either. It's perfect for people who don't want too much of any particular characteristic. It's an all-around versatile combo. If you don't mind the looks, give one a try. If you demand the best in any category, you probably won't find it in a crossover.