string = "Hello 12345 World"
numbers = [x for x in string if x.isdigit()]
print numbers>> ['1', '2', '3', '4', '5']

Another example:

>>> noprimes = [j for i in range(2, 8) for j in range(i*2, 50, i)]
>>> primes = [x for x in range(2, 50) if x not in noprimes]
>>> print primes
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]

What is confusing me is the x for x part. I've seen it a few places before but unsure what it means. Could someone provide more examples and explanation for these examples?

These constructs are called list comprehension.

Simple example

[x for x in iterable] literally means "for every element in iterable, add that element to newly created list" and is equivalent to

new_list = []
for x in iterable:new_list.append(x)

Line-by-line explanation:

Loop                             List comprehension
--------------------------------------------------------------------------
new_list = []                    [x for x in iterable]
for x in iterable:               [x for x in iterable]new_list.append(x)           [x for x in iterable]

A bit more complex example

[int(x) for x in string if x.isdigit()] can be read as "for every element in iterable, add the result of applying int to the element to newly created list if x.isdigit()".

Line-by-line explanation:

Loop                             List comprehension
-------------------------------------------------------------------------
numbers = []                     [int(x) for x in string if x.isdigit()]
for x in string:                 [int(x) for x in string if x.isdigit()]if x.isdigit():              [int(x) for x in string if x.isdigit()]numbers.append(int(x))   [int(x) for x in string if x.isdigit()]