Mine is always fairly easy. For my actual show-y tail extension, I seperate the tail and do a smal braid in the middle (Where no one can see it) then slip the tail through that, and tie up the braid or band it so it holds it in. It's not fallen out for me yet. I just figured it out myself though there might be a better way, you could try youtube.
I have smaller, colored tail extensions that just have clippies. LOVE the clippies, but I doubt any good extensions will have little teeny tiny clips, haha. But what do I know. :)
It depends on what kind of tail you have. If you have a loop attachment like I do, then you take a section of hair (under all the rest. You can put it next to the tail bone if the fake tail is used for thickening up the real one, but if you horse only has a half tail length then I like to tie the fake one in at the very end tip of the tail bone) and split it up into 3 sections for braiding. Slip one of the gathers of hair through the loop and then braid the rest as you normally would. Make sure the braid is tight, but not so tight that it irritates your horse. You can then use clips or even electrical tape for added security in making sure that the tail doesn't come out.
Also make sure to wash and condition the fake tail like you would a real one!!! :)
The method my friend uses is to do three small braids in the middle of the tail slide the middle braid through the loop and then braid the three smaller braids together to hold it in. Secure it with several braiding bands and then brush the whole thing out to blend it all together.