[Other answers cover the meaning of the lens name]
Generally, a longer focal length is required to get more zoom. But it is worth mentioning that the maximum (and minimum) focal length of many zoom lenses varies with focal distance. Focusing close will produce a different maximum and/or minimum focal length than focusing at infinity. This is simply a tradeoff in the optical formula of a particular lens.
So when fully zoomed in, your 70-300mm lens may only be 300mm sometimes.
Practically speaking there are three methods to get more pixels on the subject.
The first is to use a longer lens. It is worth noting that here the inverse square law works in your favor. A 600mm lens puts 4x the number of pixels on the subject as a 300mm lens...a 420mm lens would put 2x as many. Generally, this is the "throw money at the problem" solution. With birds, no matter how much money you throw at longer lenses, you will always want more.
The second way of putting more pixels on the subject is to use a higher resolution camera. This is also throwing money at the problem, and only scales linearly -- a 24 mega-pixel camera puts twice as many pixels on the subject as a 12 mega-pixel camera.
Finally, the inexpensive way of putting more pixels on the subject is getting closer. With birds, that comes down to field craft and field craft is a matter of time and patience. Here again, you have the benefit of the inverse square law: cutting the distance between you and the subject in half puts 4x the number of pixels on it.
To put it another way, no matter what lens and camera you have, field craft will be beneficial when trying to photograph birds. Getting close with a long lens on a high resolution camera is the best way to put pixels on the subject. The getting close is the only part that you can practice.
It is also worth noting, that using a tripod will tend to provide sharper images. A tripod is probably the cheapest piece of gear that will improve your pictures. Using a tripod is also a sound piece of field craft.