A Naive algorithm is usually the most obvious solution when one is asked a problem. It may not be a smart algorithm but will probably get the job done (…eventually.)
Eg. Trying to search for an element in a sorted array.
A Naive algorithm would be to use a Linear Search.
A Not-So Naive Solution would be to use the Binary Search.
A better example, would be in case of substring search Naive Algorithm is far less efficient than Boyer–Moore or Knuth–Morris–Pratt Algorithm
A Closed Form Solution is a simple Solution that works instantly without any loops,functions etc..
Eg:
Iterative Algorithm for sum of integer from 1 to n
s= 0
for i in 1 to n
s = s + i
end for
print s
Closed Form (for the same problem)
s = n * (n + 1 ) /2