There's pretty much every kind of transport you can think of in Europe. It's relatively small size makes it the ideal continent to travel across, be it by car, rail, plane, boat, or catapult. Okay, so there probably aren't catapults.
One of the most popular means of backpacking in Europe is Interrailing. This is essentially a single ticket that gives you free reign over most European rail services for a whole month, so you can hop on and hop off to your heart's delight. An Interrail pass is valid in *deep breath* Austria, Belgium, Bosnia-Herzegovina, Bulgaria, Croatia, Czech Republic, Denmark, FYR Macedonia, Finland, France, Germany, Great Britain, Greece, Hungary, Ireland, Italy, Luxembourg, Montenegro, Netherlands, Norway, Poland, Portugal, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, and Turkey.
So chances are you won't be squeezing all those into a month. An Interrail pass is great value if you're planning on visiting multiple destinations, but if you're only going to 2-3 places it can work out cheaper just to buy individual tickets for those journeys. It's worth doing your homework beforehand.
Practically every country in Europe has several international airports that make it easy to pick a starting point for your gap year. A popular option is flying into the UK or France and working your way east. It's also easy to travel between countries by air if you're in a hurry, though this can prove expensive. There are a number of budget airlines in Europe, but you've got a hefty backpack you'll often end up paying a big surcharge to take it with you. Again, it's worth doing some research before you travel.
Road networks across Europe are extensive and well-maintained, making a roadtrip across the continent an excellent option. It gives you absolute freedom and is a brilliant way to see the scenery. Taking a car into major cities can prove trickier, as different cities will have different regulations for parking, etc.
You can rent a car at most airports and within major cities. If you plan to take the car into different countries, make sure to approve this with your provider first. Some rental companies will not allow it. For example, a car rented in Germany usually can't be driven into Poland, but vice versa is okay.
This can be a cheaper option for moving around cities, or even going from city or country to another. There are plenty of providers across Europe, and you can often just turn up and buy a ticket on the day. Keep in mind that travelling by bus will take longer than most other methods, so make sure to plan your itinerary accordingly.
Obviously this isn't possible in many areas of Europe, but it's a very viable option in the north and south, where ferries and smaller boats are a great way to move between countries and see the beautiful coastline.