Sometimes I like to make rather high-speed descents (during which I also make jumps) in the forest near to my house. I want to know how this kind of activity is correctly called. Is it downhill, freeride, etc.?
From your description it's hard to say exactly what kind of riding you are doing, all forms of mountain biking can involve going down hills quickly and jumping. If you want to be specific it sounds like it fits into one of these categories -
Trail Riding This is a good catchall term for recreational mountain biking where the trail is the most important thing - It includes climbing and descending, typically on singletrack. It's possible your wheels will get off the ground, but not mandatory.
All Mountain Again this includes climbing, but all mountain riding focuses more on going down than getting up. A all mountain bike will generally be heavier and more robust than a trail bike - it needs to handle higher speeds and bigger impacts.
Downhill Are you just interested in getting down that hill as quickly as possible? Are you pushing/getting carried to the top? Then you're riding downhill. You'll probably be jumping, and you'll probably be going very fast.
Freestyle Like downhill freestyle is all about the descent, but in this case style is more important than speed. If you're riding freestyle you'll be doing tricks and stunts on the jumps and drops.
Bear in mind that these aren't strictly defined terms, and are prone to change.
Yep, there is a name for it and in the UK (and presumably Canada) it is known as 'North Shore'. This is a shorthand for British Columbia riding style on trails exactly as you describe. Even the bike manufacturers have cottoned onto the trend, plus the editors of Wikipedia:
Now you know the magic YouTube search term, find some good videos to post here for the enjoyment of the stackexchange masses. Good luck with the trail building and, at the risk of sounding like your mother, don't forget your helmet and body armour!
"Freeride" generally refers to the type of riding described above, using Summertime ski-trails to ride essentially downhill. You usually truck up to the top.
"Downhill" normally refers to a particular type of MTB racing on a downhill course.