For PvE the best options are the Life Ring, the Fire Gem or the Sovereignless Soul. Everything else is either easily obtainable fairly early on, plentiful later in the game, or nearly useless until it can be obtained mid or late game.

What is the best burial gift for a knight in Dark Souls 3?

The Life Ring is the best starting Burial gift in Dark Souls 3. When you start in Dark Souls 3 you don't have a lot of health and that bit extra health would help you in many scenarios you can think of.

What do the burial gifts do in Dark Souls 3?

It is a consumable item that adds 2000 souls to your currency pool once used. If you decide to select the burial gift, I suggest using it only when you need it, as it's always protected in your inventory. This consumable is especially useful if you plan on leveling up early on in the game.