Maple seeds (called "samaras") are best harvested when the papery wing part of the seed turns brown, but the thick fleshy part of the seed is still slightly green and not dried out. They need to be stratified, which means they must endure a period of cold. What I do is harvest the seed, break the pair of seeds in two, and then plant them in a shallow 4" garden pot - cheap plastic will do. Do not plant them deeply. You may want to cover the pot with some wire screening to keep out the squrrels and chipnumks if you have a problem with those in your area. Plant as many pots as you want and then find an area for them to spend the winter. They need to be exposed to the cold and rain. In the spring you should have new maple seedlings come up. If you don't see seedlings, don't give up. Sometimes it takes 2 years for them to sprout. Then just transplant the baby trees to whatever part of your garden you wish. Easy!

P.S. They can grow for a whole year in that 4" pot if necessary.

Thanks for the responses! How often do maples get seeds? I just moved into the house last summer and the maple didn't have any seeds at all last summer. This year there are a ton, though!

Spring weather during flowering does affect seed volumes but generally mature maples set a lot of seed every year.

The above info is very good but before you proceed, you should know one very important point. The seed you plant from your beautiful tree has a very good chance of looking nothing like the parent tree. It's a total gamble as to what the baby tree will look like. It's just like when two dark haired parents have a blonde child. You never know what kind of genetics lie deep within the seeds DNA. If you truly want a "clone" of a certain tree, it must be grafted. You can try this yourself or purchase a grafted young tree.
If you're determined to plant a seed, I've also had good luck putting so
e seeds in moist peat moss in the refridgerator (just above freezing). Put them in after harvest in the fall and bring them out 3-4 months later in early spring. Plant the young shoots in a good potting soil until established.

Actually with sugar maple, your chances are pretty good that it will look similar to the parent. There will be some variation. Some will have more orange or more red in fall color. Some will break bud a few days later. Some will have somewhat narrower or wider branch angles, but overall they will say, "I'm a sugar maple!"

This is NOT true if you have a grafted tree selected for some uncommon mix of traits. E.g. Crimson King norway maple is a genetic sport with red leaves. These sorts of traits are usually a combination of two recessive genes. The F1 offspring will rarely show the trait. But I would expect offspring of something like Unity or Inferno Sugar Maple (both chosen for fall colour) to have offspring between the parents and normal seed run maples.

(Apples on the other hand have a truly wild and wonderful genome. Cross two apples. Plant 1000 seeds. Maybe 100 will resemble ONE of the parents in some trait. Apple breeding is lots of planting and ruthless culling.)