It's essentially a sandbox. You start up the game (and I'm going to use Creatures 3 as a reference point, as the others are just old and would confuse the matter) and are offered two choices. One; start off with a breeding pair of adults. This will create two norns who will wander around, eat cheese, give each other wild smacking kisses, and repeatedly kick the radio until the female gets pregnant and lays an egg.
After a long amount of time left unattended (the mother utterly ignores the egg after spitting it out), the egg will grow and eventually hatch into a bouncing baby norn, who will babble nonsense while eating cheese and repeatedly kicking the radio. In order to understand your little one, you need to coax it into staying in the learning room while you run the language teacher on autopilot.
In the old versions, it took a lot of hands-on training and repetition before they got a word, now you just have to make sure they stay in the room. A cheese-dispenser is provided to help out, but it is recommended you stick a radio in there so they can kick it. They will still learn even if they are utterly distracted.
Okay, now you've got an incredibly cute baby norn that can now express its thoughts and feelings. You can now take on a hands-on approach to teaching it, or you can let it figure out stuff by itself.
Norns are born without any knowledge of their surroundings. They have individual personalities, and some will bravely venture forth and interact with everything that they see in an attempt to figure out what it is and what it does. This starts out with just hitting the object, but as a norn explores its own capabilities they will eventually push buttons, pick up objects, and try to eat inorganic material.
Their actions determine their future behavior, and so if they are left to their own devices they will develop habits. Very strange habits, most of the time.
However, you also have an effect on norns. You can lead them around by the hand (and if you manage to get them to work with you on getting one of the secrets, you can later pick them up and move them around more easily), tickle them, or slap them. Your opinion of what they do has a very strong effect on their future actions, so you can quite easily train them to not constantly push the door control button, causing them to continually pop in and out of the corridor.
You can set any number of goals for yourself. Are you going to raise a large number of norns who will viciously punch the other inhabitants of the spaceship, so you can suppress the other life forms by force? Are you going to nurture a small group of norns so that they not only have the ability to complete a myriad of complex tasks, but will also teach these things to their young?
Another big thing about Creatures is the incredibly complex genetics system. It is possible to specifically breed norns to have certain traits, and although I was never much of a breeder I assume that they will at times have generations that contain mutations to the DNA. You can, after a lot of hard work and time spent at the gene lab, create a strain of norns that are highly resistant -if not completely immune- to disease.
One of the most sought-after traits is the ability to breathe underwater. However, I don't think it is actually possible outside of a few special norns created specifically for Creatures 1.
But yes, they're all cute. Even the "evil" grendels are cute. And you can actually get quite attached to a particular creature, and be rather distraught when he finally can't live any longer.
On top of all this, there are also fun things to be had with various machine parts around the ship. With a couple clicks, you can hook a wire from the creature-sensing pressure plate to the door, causing it to open whenever something steps on the plate (I was using this as a defense against hostile critters until one of the other alien lifeforms stole the pressure plate before it got booted out).
I once made a hostile-destroyer, which was this floating platform that would hover around until it detected the lifeform it was geared for, whereupon it would sound an alarm, follow after it, and fire mud pellets from the closest thing the game had to an actual gun (it could kill creatures, however).
So, eh, yeah... There's more to it, but I could go on for ages.