basically in a hydro system you are feeding the plants directly when they are watered rather then in soil where the plant absorbs food out of the soil. Hydro can be very complicated and I recommend researching alot and buying the proper equipment before you start..ie ph/ppm meters, ph up and down, a full set of nutrients and of course the system itself just to name a few. A major advantage of hydro is faster growth rates and more control over your plants, and automated watering can be very nice too. One of the reasons for faster growth rates is the fact that your roots are getting alot of oxygen...roots need oxygen just as much as they need air. Some the best hydro systems are aeroponics, where the roots are basically hanging in air and sprayed at regular intervals with a mist of nutrients, and allowed to get plenty of "air" between watering cycles.
As far as what to "replace" the soil with - you can replace it with nothing and just have the root system hanging freely in an aero system..or you could add some hydroton..which are basicaly little clay pellets that the root structure can form around, and also hold a bit of moisture...you can use rockwool, lavarocks, I've seen people use tons of different mediums to grow in when it comes to hydro. A hydro medium should have a nice neutral ph and not have any nutrients in it...or maybe slight trace elements, you don't want nutriants in your medium because you will be adding them yourself, and it's important to know exactally what is going into your plants.
Just start reading up on some of the hydro faq's and look through a few people's grows. Hydro can be quite a bit more expensive to start out then soil, and can be difficult to get the hang of if you don't stay on top of it. Making a mistake in hydro can easily kill your plants if you don't recognize it right away, where as in soil you usually have a few days to adjust. but all in all hydro is definately worth it if you take the time to make sure you know what you are doing before you start.....imo you can kind of "crash course" through soil for the first time...but not with hydro...you might want to look into coco as well...it's kind of described as "hydro results with the ease of soil". I prefer coco over soil and hydro, it's kind of in between, but you still have to know your nutrients and ph well.
also i highly recommend building a hydro system yourself if you are capable or good at building stuff..prebuilt ones can get really expensive. No matter which one you buy..in 6 months you'll be saying to yourself "I could have built this myself for 1/3 the price." but some people either can't build stuff or don't realize how simple these systems are until they've been around one for a while.
I'm not an expert in anything..take this all with a grain of salt and get a second opinion
