Spaying your female cat before her first oestrous cycle (going into "heat" or being able to breed) greatly reduces risk of cervical cancer and eliminates the risk for ovarian cancer, because removing the ovaries reduces the levels of hormones that encourage the growth of cancerous tumours. Spaying reduces your cat's risk of mammary cancer as well.
Keep in mind that there are other diseases, resulting from natural cat behaviour when they mate. Feline leukaemia and feline AIDS are two diseases spread through infected cats biting other cats, (these diseases are different from human AIDS and leukaemia, and cannot be transmitted from cats to people). By reducing your cat's urge to fight over mates and territory, you'll also reduce their chances of contracting these incurable diseases from other cats.
Unneutered male cats are driven by hormones to seek mates and defend their territory against intruders. So, two or more unneutered male cats in the same household can spell trouble. Fights tend to break out, especially if there's a female cat in heat nearby. By neutering your cats, you'll reduce their aggressive instincts.
Reduces Risk of Roaming
When female cats go into heat, both her hormones and instincts are urging her to find a mate. If she's your only cat, she'll try to escape every time you open the door so that she can find one. Remember that males are also driven by hormones and the mating instinct, and will try their best to escape for the same reason. Both males and females are at risk outdoors of being injured e.g. crossing roads. By neutering your cat, you'll reduce this wanderlust and find they're happy to stay put in the safe, comfy spot next to you on the couch.
Male cats spray their urine on vertical surfaces to mark their territory. The pungent odour of an unneutered cat's urine alerts other males that there's another guy nearby who has claimed the area as his turf and it tells females he's waiting for his opportunity to mate with her. An unneutered male cat in your house can be a messy business. Neutering a cat reduces or eliminates the urge to spray, and if they do, the scent should be much more mild.
Female cats also pass bodily fluids when they go into heat. These fluids also contain scents to alert males that a fertile female is nearby. By spaying your female cat, you'll eliminate the same problem.