For one year olds
Melissa and Doug -
They make quality wooden toys perfect for this age group, and can usually be found on sale on Amazon. This car carrier has been a hit with every kid I've ever met. They also have play sets with a vehicle and several figurines. The tractor and firetruck were most popular at our house.
Chunky puzzles are great simple shapes that also have a picture of the piece for easier matching. The pieces also stand up on their own, for creative play.
Active toys -
A pop-up tunnel is good for outside or the living room on a rainy day.
Bilibo looks like nonsense, but it is a fantastic open ended toy for this age - and up!
Little Tikes makes great ride-ons. Our daycare had this one and it was fabulous. At home we had the Prince Lionheart Wheely Bug, and it was fun, but is more difficult because of its construction.
Now that they can somewhat feed themselves, these snack catchers are great - especially for the stroller or car. Out to eat at a restaurant can be challenging, so we always took our own plasticware (these have a carry case!) and a waterbottle (since some places don't have cups with straws/lids).