The HIVE witnesses are basically those who keep the HIVE blockchain running smoothly. They run 'nodes' that monitor all of the transactions on HIVE, ensuring they are valid transactions and that there are no duplicate transactions. I do not know all the technical ins and outs of how 'nodes' work, but I know the top witnesses put a lot of time and money into the ongoing monitoring of the blockchain.
Delegating is a bit easier to explain. If you have HIVE Power (HP) in your wallet, it increases what you can do on the HIVE blockchain. The higher the amount of HP that you have, the more frequently you can post, comment and upvote content. It also increases the value of your upvote. So HP is very valuable on the HIVE blockchain.
So there is a built in function in your HIVE wallet where if you have excess HP to what your posting, commenting and upvoting needs are, you can choose to delegate (similar to a loan, but without the risk) some of your HP to a HIVE user account to boost up their HP.
As a reward for your delegation that user account may pay you a regular daily amount in HIVE (like an interest payment). Only certain accounts do this, so choose wisely who you delegate too. The HIVE payment will continue daily as long as your HP remains delegated.
If after a few weeks you decide that you want to post more, comment more or upvote more and you need more HIVE Power in order to do so, you are able to remove your delegation and get the delegated HP back in your account again.
It is a very useful feature on HIVE, especially if you do not post regularly, but want your HP to be working for you and earning more HIVE.
Hope this all makes sense.