Can I Grow A Fruit Tree From Seed?

Yes, you can grow a fruit tree from a seed or stone obtained from fruit in the supermarket, but you will not get the same variety. You will often get a random mongrel tree with inferior fruit.

• This phenomenon is referred to as "not coming true" from seed, meaning that the offspring are typically very different to their parent.
• This is in contrast to a wild, uncultivated tree, most of whose offspring will vary only slightly. 
• For example, you have a Cox's Orange Pippin apple tree.
• That tree will only produce Cox apples, regardless of which apple pollinates it.
• When you grow its seeds, you will never ever get a Cox apple tree.
• Most seedlings produce poor quality fruit, but some will produce a tasty new variety.
• The number of attempts required varies a lot, but with the correct approach, it's possible to grow more seedlings with fruit worth eating than not. See the playlist at the bottom of this page.

There is no shortage of whimsically charming but woefully misleading videos like this, falsely telling viewers "don’t throw out your apple seeds, grow a honeycrisp plant instead"

No, you can't grow a honeycrisp apple tree from a honeycrisp apple seed!

So how are new fruit trees made if not from seed?

New fruit trees of a given variety are grafted, which effectively clones them. The cutting of the variety you want is called the scion, which is grafted onto a rootstock, which is a different apple tree variety.

Are there fruit trees that do come true from seed?

Yes, but not any of the cultivated fruit trees that are traditionally grown in the UK. 

If you are growing vegetables, the distinction is usually easy: hybrids do not come true from seed, open-pollinated varieties (i.e. all heirlooms) do come true.

Can I breed fruit trees from seed at home?

Absolutely, if you have the space and patience. By hand-pollinating your trees and protecting the flowers from bees, you can choose which trees to cross, which will increase your chances of getting a new variety that you like.

Don't believe the sources that tell you it's necessary to grow 10,000 apple seedlings to find one good new variety. They are giving you a worst case scenario, and by "good new variety", they don't mean "worth eating", they mean "good enough for a breeder to patent and grow commercially to compete with existing, well-known varieties that everyone is comfortable buying."
By that high standard, a delicious, disease resistant apple with ugly looking skin is a failure, because shoppers primarily buy fruit based on how it looks. 

Here's what a commercial apple breeding program looks like:

That's not going to work for the average gardener, so over to self-taught expert home orchardist, Skill Cult Steve: