Pleasure purring: In general purring has been associated as an expression of pleasure, or also as a form of communication with their young. Often a contented cat will purr on an owner's lap, and usually that means you have a happy cat at the moment!
Self-healing purrs: Something puzzling is that cats also sometimes purr under stress or when recovering from an injury. Scientists are beginning to find that this is because purring seems to actually improve their healing process! Various studies showed an improvement in bone density during purring and it also seemed to promote overall healing.
Kitty kiss: If a cat is in a relaxed mood and it gives you a blink - it slowly closes the eyes and then opens them, then you have been kitty kissed! According to many cat owners, this is the equivalent of a friendly cat greeting, smile, nod or kiss!
Half-closed happiness: When cats are relaxed and content, often they will close their eyes partway. Generally if a cat is on your lap with half-closed eyes, that means the cat is feeling relaxed and content and maybe a little sleepy too - but overall, it means the cat most likely likes the way you are petting!
Tummy rubs = trust: A cat's underside - the chest and belly are vulnerable areas they want to protect, and in defense a cat may lay back in order to defend better with all four paws. If a cat is lying back many cats will attack your hand if you attempt to pet an exposed belly... however if they allow you to, that conveys happiness, comfort and that they trust you to be gentle and kind with a vulnerable part of themselves.
Love licking: Cats lick each other to groom each other as well as bond with other cats they know very well. Cats also lick their owners and other people for similar reasons - to 'groom' them, and show care and affection. So if your cat, or another person's cat decides to lick your hand, that doesn't mean you smell like tunafish - it means they may like you!
Sandpaper tongues: Many cat owners like it when their cats lick them, but not for too long - cat tongues are very rough, like sandpaper! The little jagged bumps help catch loose hair, skin and fleas (if they have any!)
You belong to your cat: A cat will often rub along an arm or leg of a person. This could have many meanings... it could be a way to attract them for petting, playing, food or attention. It could also be that they are 'marking' the person. Cats have scent glands around the mouth that they will subtly use to mark the person as their territory.
Cats rub each other too, as a friendly greeting and to mark each other for easy recognition in the future! They also rub on furniture, trees, an all kinds of objects... that means they are marking all these items... marking it "mine!"