Whether its football, rugby, cricket, tennis or golf, sporting events and matches are usually no short of drama, and the players can be the subjects of some pretty spectacular photographs if you're a budding sports photographer, and the staff at the venue don't have issue with you bringing along a hefty camera and a lens.
Obviously now your lens choice will depend on the sort of sport it is, but for most people they will want to 'freeze' the action, and not have much or any motion blur in the images at all. This at its heart will require a fast shutter speed. For events during the day this is not usually too much of an issue, as the abundance of ambient light means you can raise the shutter speed and lower the ISO for clean and crisp images. You still need the reach, however, so we're talking about at least a 100mm focal length, if not much longer, depending on where you're sat! For indoor sports you won't need to go too long, but you will need a fast lens with a large maximum aperture. Indoor events cut out the sunlight, and artificial lighting is usually nowhere near as powerful, so you'll need to compensate.
The issue with longer focal length lenses, is that they don't often have the same wide apertures as shorter focal length lenses, and certainly not for those on a budget. You can get hold of some good 200mm ish lenses with maximum apertures of around f/2 - f/2.8 for a reasonable price - but look at the ones with some sort of image stabilisation if you can, to eliminate a degree of camera shake as they are heavy bits of kit, and at 200mm any movement by you will be greatly exaggerated.
As I said longer lenses do exist, in the 400 - 600mm range, and these are the sort of ones used by the press pool at the front on the sidelines, but these might set you back anywhere from £4,000 to £12,000. You can cheat, however, by using what's called an extender. They often come in a few different variants depending on your brand, with perhaps a 1.4x and a 2x extender. As you may have guessed, these will (by placing between your camera and lens) extend the maximum focal length of your lens. So if you have say a 70-200mm lens, you now have with a 2x extender a 140-400mm lens. This doubling will also affect your maximum aperture as well, so you now only have f/5.6 to play with, but given adequate light, this is still usable if you bump up your ISO a little.
So really your lens choice will depend on how fast paced the sport is, and primarily whether it's an indoor or outdoor event and how fast back from the action you maybe sat/stood. For indoor you will benefit from higher maximum apertures but no more than 200mm focal length really, so maybe the 135mm f/2 or 200mm, or even the 70-200mm f/2.8. For outdoor you will probably be further away, so having extra reach is handy, a 70-200mm with an extender, or if you can stretch to it, a 400mm. Of course primes limit your versatility in framing, as you will not likely be able to move around much, so a zoom is perhaps more versatile, albeit they won't let in as much light so you may have to compensate.
Although this series, and this post, focuses more on the lens choices, the camera too plays an important role. For example both Nikon and Canon have a line of high frame rate full frame and crop factor bodies that are incredibly useful for this sort of discipline. Remember that crop factor bodies will have a built in 'extender' for your lenses, giving you 1.5x or 1.6x (depending on brand) multiplier for the focal length, albeit with a bit of a loss of capability and sharpness in low light etc.
Informative posts about weddings and related things, as well as general photography stuff.