Begin by learning these two (explained and illustrated by the skipper)
The Molly-cover-securing knot (is only two half-hitches)
The Waterman’s/Bargee’s/Lighterman’s hitch (ideal for our stern line)

Now the three most useful knots for ordinary boaty use. They will enable you to look quite professional in almost any knotty situation:-
Round-turn-and-two-half-hitches
Bowline
Reef knot

When you have mastered those, rather like "Buy-One-Get-One-free", you will have already inadvertently learned these!
Half hitch
Clove hitch
Overhand knot

Ready to extend your repertoire?
Go for:-
Figure of eight
Rolling hitch
Sheet bend

In some situations, it is useful to tie knots “on-the-bight”. For example "Round-turn-and-two-half-hitches on the bight".
A bight is just a loop; sometimes a rather long one.
It is a way of using up excess lengths of line, and is useful when the end of the line is not accessible, or when it would take too long to thread it around (such as when securing the bow line at our mooring).

An anonymous contributor kindly showed me this excellent website - Animated Knots - Use it in conjuntion with my list on the left

Some other perspectives of popular knots:-
Two Half hitches
Bowline (pronounced Boh-Lin)
Reef knot
Half hitch
Clove hitch
Overhand
Figure eight
Rolling hitch
Sheet Bend

Simpler illustrations of some of the above

Animated versions of some of the above
Other animated versions of them

Hungry for more?
Loads of knotty links
Want to spend some money?
The knot-tiers bible is “Ashley's book of knots".
On a budget? - Browse through a few library books, or a book shop - Choose any uncomplicated book with clear illustrations and comprehensible instructions.