Cancellation is when a stamp is used and the post office applies a mark (stamp), most frequently but not exclusively in black, to cancel the stamp. This prevents re-usage. Often it also shows where and when the item was posted. Some cancellations also tell you how the stamp was used: e.g., postage, in payment of tax, telegraph, railway carriage, etc.
Mounted and unmounted are terms frequently used in Great Britain. They are the same as hinged and never hinged, respectively. These are used in connection with mint stamps. The hinge disturbs the gum resulting in a stamp that is no longer as sold, i.e., post office fresh.
Some sites you mention run auctions. Auctions may post pictures of stamps that are varieties. They may look the same. Experience often helps to recognise the varieties.
I think the remark that hundreds of rare stamps from the era you mentioned is not so much a remark about value. There are many sites that say "rare" stamp when the average dealer will have more of them in stock than there are fish in the sea.
ebay is one of the sites where all types of crackpots will claim rare stamps or sets, when they even are incapable of recognising the stamps. Not many stamps from that era are rare and they do not tend to come by the hundreds. Also, if a site posts a variety, that may be rare but the stamp without that variety may be awfully common.
If you would consider putting them in the bin, some little boy or girl might be extremely happy with them, probably more so than your bin.
Before doing so, it might be useful to post some pictures here. There are many people that have experience in one or more areas – hence the advice to sort by country – and will tell you whether it makes sense to have a further look at stamps. They will know what stamps indeed are rare or have rare varieties.