The correct forms are "had better (do something)" and "had best (do something)".
[ For historical nuts, the "had" is a calcified past subjunctive of the modal verb "to have to", and denotes that the doing of something is not real but strongly suggested. The past subjunctive once carried a conditional meaning, so it the language of Shakespeare and his contemporaries "it were" can mean "it would be". Thus "he had better run away" = "it would be better that he should run away" and "he had best run away" = "it would be best that he should run away". ]
The form without "had" is strongly colloquial or illiterate, in the same way as the "have" in "have got" is sometimes omitted, yielding "I got to" or "I gotta". "I better" is common enough, "I best" does not sound natural, but even "I better" should not be used in any kind of formal setting and NEVER in writing.
"He had better do something" indicates a course of action that would improve the situation, make it better.
"He had best do something" indicates a course of action that would improve the situation to its maximum or best extent.
"He had better do X" leaves open the possibility that doing Y might be even better, though X is preferable to what is going on now.
"He had best do X" means that nothing will be as good as doing X, even though X will not necessarily resolve all the difficulties of the present situation.
PS. Another usage of the same idiom, one in which the subjunctive/conditional meaning is very clear, is
I had rather you go there
I would rather have you go there = I would compel you to go there, instead of what you are doing now
and is stronger than
I would rather you go there
I'd rather like you to go there = I would prefer you to there, instead of what you are doing now.