That's the first question anyone asks when they're interested in reloading, but it misses the point. It's easy to save money on ammo: just don't shoot! If you're getting into reloading just to save money, you're doing it for the wrong reason. There are several good reasons for reloading that have little to do with cost: 1) You can make ammo that's more accurate than factory rounds. 2) You can buy better hunting bullets (e.g Barnes, Nosler Partition) than are available in most factory ammo. 3) You can tailor your ammo to your gun or to your specific needs, and 4) Since the per-round cost is less, you can practice more for the same money and become a better shooter.
The actual cost difference varies a great deal depending on the particular cartridge and what bullets you use. For many of the big magnums, the factory ammo is so expensive it would be insane not to reload. Of course, most people can't shoot too many rounds of .338 Winchester Mag in a session, so you don't go through too much ammo with the big magnums. On the other hand, 9mm Luger ammo can be purchased dirt cheap, so there's not a lot of cost savings per round to reload, but most shooters will can go through several hundred rounds in a day, so the savings add up.
Generally speaking, it costs less than half the price of a factory round to reload it. But that assumes you have the brass. I don't like to use factory ammo, so I buy new brass. Consequently the first time I load new brass it usually costs more than factory ammo. The exception for me is .17 Remington Fireball; this ammo is so absurdly expensive that I can load new brass cheaper than I can buy factory ammo. So it all depends.
Let's take the cost of reloading a popular cartridge like .308 Winchester. I'm going to use the Winchester 150 grain Power Point bullet, since it's available to reloaders. In fact, most reloaders opt for "premium" bullets that are more expensive than these.
New ammo is available from Midway USA for 18.39/20, or 92 cents/round.
To reload, assuming you have brass:
150 gr Win PP bullet: 25.75/100 = 26 cents
Winchester LR primer: 26.00/1000 = 3 cents
45 grains IM-4064 powder 19.95/lb = 13 cents
TOTAL: 42 cents/round
So it costs roughly half as much to reload a 308 round as to buy it new. If your time is worth anything, you may come to the conclusion it's not worthwhile to reload. But, as I said in the beginning, saving money is the wrong reason for reloading. I prefer to use the term "handloading", because you're really crafting ammunition to your needs.
By the way, my reloading component costs were from Powder Valley. They don't include shipping and hazmat costs, but if you get together with some of your friends to order in quantity, you can get those costs down substantially. (You pay one hazmat charge for powder and primers, regardless of how much you order, up to 48 pounds.)
If you are interested in getting into handloading, see if you can find a friend who loads and get him to help you load a batch of ammo on his equipment. You might have to spring for a set of dies for your cartridge, but most reloaders are happy to help someone get started. If you decide it's not your cup of tea, then you haven't invested much. Better than springing for several hundred dollars worth of equipment only to find you don't enjoy it.