Hello MayMac,
No your topic title is clear, and I think I will give you some useful answers.
Actually you can make video hosting even using your own server, the issue is you’ll probably meet hard bandwith issues (depends your server accounts, but you risk to meet hard bandwith issues).
That’s ok if your website doesn’t do a lot of traffic and you don’t post vids of 1 Gb each (compression helps a lot on that matter, mp4 so on), but still.
After, on the other hand if you plan to make a lot of videos – tutorials on how to garden, make up for girls, or let’s play of games in high hd etccccccc – you’ll probably soon hit a free account quota.
Then I’ll advice you JwPlayer, this is also a WordPress plugin, that can scale to your audience. It offers you a good player (responsive, flash or not, so on). And if you see your videos are bumping it will cloud and work. I precise I’m not affiliated with them, just they make a great WP plugin with a free version (you can choose to use JW as free but you quote their brand, and will block at some limits). But to me it’s a good start.
As a conclusion with WP and videos, they are tons of solutions, I just think JW Player is a good one to start with (you host videos on your server and use the player, then if it scales you can fully transfer to their cloud).
After, if you have no problem your video can be accessed with no restrictions, YouTube is now nice, no restrisction so on like many vids publishers, but it’s a completely other story (B2C, B2B, number of hits per month, so on).
Hope my 2 cents, helps,
NB: I don’t work for any cloud hosting provider, just quoting some solutions I think are good. The industry is changing so fast, it’s hard to tell but any cloud is good, to benefit from scale effects and their global CDN.