I've decided to share a few things we do right now which seem to be interesting and fun and (I think) help her develop (or at least don't slow her development:). Just an explaination: I write her all the time because I have a daughter. At this age it doesn't change much for boys I guess.
Activities with children's books
There are many books for toddlers: thick "pages", little text, but many colorful and varied pictures. Turning pages in such a book seems to develop motor skills, pointing and naming various things teaches some words, then asking "where is < the thing I just showed you >" and waiting for the baby to point seems to be fun too, and it is a good opportunity to praise the child. Picking up the book, putting it down and getting another one, then reading and pointing and naming and asking again - it can go on for long minutes. Of course simply reading the text and pointing at things you read about also works.
Picture albums work similarly as books, but instead of naming things you are naming people (and, perhaps, things). It's a great idea to have a special album for the child and have pictures of mom, dad, siblings, grandparents and other family members or friends.
Some household equipment can be operated by the baby. There're buttons to press, knobs to turn, possibly even some levers, and many of them cause some diodes to blink, messages to display, etc etc. A stereo works great for my kid, she presses a button and a drawer will open, another button toggles a led under it, and yet another makes a casette deck open. For some reason it's all fun.
Of course normal things which we use everyday are also interesting. A wooden roller, a phone charger, a camera case, a bag... Virtually everything.
Provided it's safe. Make sure there are no suffocation dangers, no wires exposed, no unprotected plugs, no sharp edges, etc. And keep an eye on the baby when it first plays with something new. There're ways to harm oneself for a toddler we couldn't have imagined.
Apparently it's a lot of fun to put various things into containers. And get them out of containers. And shake the containers with something inside. And do many other things I'd consider boring. Just make sure the container has no sharp edges (for example some formula cans do, some don't). My LO seems to be slowly grasping the concept of size, firmness, things fitting in each other and so on.
Discovering the world
When the baby learns crawling (and, later on, walking) it gains a completely new way of discovering the world. Make sure to let the baby do it. Try to reduce the amount of dangerous things at your home (take a look at the baby proofing the aprartment question). Also, try to maximize the safe zone which your baby can explore. Put all dangerous things in one or place instead of scattering them around different closets. That way, the safe closets will be available for opening for the baby.
Movement: sit-ups, squats, swinging, dancing
Obviously the baby is not capable of doing those exercises, but she seems to enjoy when I do them. Pickaboo sit-ups, squats while carrying the baby, some swinging and even dancing (waltz seems to work fine) - all combined with making a lot of sounds and laughing - seem to be very enjoyable. Just make sure to have enough space and be safe!
There's a lot to do outdoors. But the general rule if you go on a walk should be: let the child decide what to do. If she wants to stare at ants for fifteen minutes - let her. If she wants to watch the ducks, chase leaves, run hands through sand, do anything that isn't dangerous - let her.
Of course, if you are on your way somewhere, that rule doesn't apply. You go out to get somewhere, not to play. Make sure to explain it.
Swimming pool is a great fun place for children of all ages. We took our LO to a swimming pool when she was around 7 months old, but it can be done earlier. Splashing water is always enjoyable, and later on there're things to throw, things to touch and in general - things to learn (how strangely things behave in water, don't they?)
I think there's no explaination needed here. ZOO is a great place to go with all those animals. It'd be best to go there everyday just to look at that funny giraffe.
Freash air, birds, trees, flowers, bushes, fungi - a lot to watch, some things to touch and a lot to learn. Just make sure no strange berries are being eaten.
A one year old is (usually) simply mastering walking, so there's not much sport to do. But running, some ball-kicking or throwing is a possibility. It depends on a child.
They're always fun. My daughter can stare at them for 10-15 minutes straight. Fascinating.
Slide and swing
Most swings and slides are not appropriate for a one year old. However, you can swing with your LO on your lap. Be extra careful not to let your child go. Slides, at least those small ones (1m?) are usable with parents' help. We lift the baby to the top of the slide, delicately hold her when sliding down (her back/head are supported by one of us) and the other parent waits at the end of the slide and smiles and laugs to the baby. A lot of fun!