In the case of Lord Sugar he was nominated by the then PM, Blair. There are hereditary lords, who get the title when their dad pops his clogs, and those nominated by political leaders.
There are two types of peerage (lord). First one is inherited by the first born son of someone whose distant relative did a King or a Queen a great favour and was knighted to be Lord Smurf of Choppingham for example. His wife would be Lady Smurf of Choppingham. The Lord gets paid for turning up to the House of Lords.
Latterly there have been Lifetime peerages which last only the life of the person who gets it. These also get paid for turning up.
You can also be knighted and become a Sir without being offered a peerage. eg Sir Jimmy Saville. Usually for good work in the community.
And a man may be a baronet, which is shown as 'Bart' after his surname (Sir A. B, Bart). That now means he is a Sir, a knight, by inheritance; his father was a baronet.
Lord Coe has things the wrong way round. He was made a Lord first, for political reasons, and made a knight later, for his services to sport. Usually the lesser honour of knight is awarded before the peerage, the higher rank, is granted.