When an email bounces, it generally means it can't be delivered to an inbox. "Hard" and "Soft" designate the two groupings of failures: one's more permanent, and one's less permanent.
Hard bounces = permanent delivery failures.
A hard bounce is an email that couldn't be delivered for permanent reasons. Maybe the email is a fake address, maybe the email domain isn't a real domain or maybe the email recipient's server won't accept emails. There are lots of reasons that an email could be a hard bounce, but the core of it is that it's a permanent failure. You should remove all of these addresses from your list.
Soft bounces = temporary deliverability failures
A soft bounce is an email that couldn't be delivered because of temporary reasons. An inbox may be full or the email might be too large and many other reasons. If they get a soft bounce on an email send, most email providers will continue to try to deliver the email over the period of a few days. You should keep an eye on these addresses -- if you notice that the same ones are popping up over and over again, it's best to remove them.
Try to keep your total bounce rate under 2% -- much higher than that, and you may start noticing some deliverability issues.