Here is my code. I'm not exactly sure if I need a counter for this to work. The answer should be 'iiii'.
def eliminate_consonants(x):
        vowels= ['a','e','i','o','u']
        vowels_found = 0
        for char in x:
            if char == vowels:
                print(char)
eliminate_consonants('mississippi')
                You can remove a character from a Python string using replace() or translate(). Both these methods replace a character or string with a given value. If an empty string is specified, the character or string you select is removed from the string without a replacement.
The line if char == vowels: is wrong. It has to be if char in vowels:. This is because you need to check if that particular character is present in the list of vowels. Apart from that you need to print(char,end = '') (in python3) to print the output as iiii all in one line.
The final program will be like
def eliminate_consonants(x):
        vowels= ['a','e','i','o','u']
        for char in x:
            if char in vowels:
                print(char,end = "")
eliminate_consonants('mississippi')
And the output will be
iiii
Using in a string
def eliminate_consonants(x):
    for char in x:
        if char in 'aeiou':
            print(char,end = "")
As simple as it looks, the statement if char in 'aeiou' checks if char is present in the string aeiou. 
A list comprehension
 ''.join([c for c in x if c in 'aeiou'])
This list comprehension will return a list that will contain the characters only if the character is in aeiou    
A generator expression
''.join(c for c in x if c in 'aeiou')
This gen exp will return a generator than will return the characters only if the character is in aeiou
Regular Expressions
You can use re.findall to discover only the vowels in your string. The code
re.findall(r'[aeiou]',"mississippi")
will return a list of vowels found in the string i.e. ['i', 'i', 'i', 'i']. So now we can use str.join and then use 
''.join(re.findall(r'[aeiou]',"mississippi"))
str.translate and maketrans
For this technique you will need to store a map which matches each of the non vowels to a None type. For this you can use string.ascii_lowecase. The code to make the map is 
str.maketrans({i:None for i in string.ascii_lowercase if i not in "aeiou"})
this will return the mapping. Do store it in a variable (here m for map)
"mississippi".translate(m)
This will remove all the non aeiou characters from the string. 
Using dict.fromkeys
You can use dict.fromkeys along with sys.maxunicode. But remember to import sys first! 
dict.fromkeys(i for i in range(sys.maxunicode+1) if chr(i) not in 'aeiou')
and now use str.translate. 
'mississippi'.translate(m)
Using bytearray
As mentioned by J.F.Sebastian in the comments below, you can create a bytearray of lower case consonants by using
non_vowels = bytearray(set(range(0x100)) - set(b'aeiou'))
Using this we can translate the word ,
'mississippi'.encode('ascii', 'ignore').translate(None, non_vowels)
which will return b'iiii'. This can easily be converted to str by using decode i.e. b'iiii'.decode("ascii"). 
Using bytes
bytes returns an bytes object and is the immutable version of bytearray. (It is Python 3 specific)
non_vowels = bytes(set(range(0x100)) - set(b'aeiou'))
Using this we can translate the word ,
'mississippi'.encode('ascii', 'ignore').translate(None, non_vowels)
which will return b'iiii'. This can easily be converted to str by using decode i.e. b'iiii'.decode("ascii"). 
python3 -m timeit -s "text = 'mississippi'*100; non_vowels = bytes(set(range(0x100)) - set(b'aeiou'))" "text.encode('ascii', 'ignore').translate(None, non_vowels).decode('ascii')"
100000 loops, best of 3: 2.88 usec per loop
python3 -m timeit -s "text = 'mississippi'*100; non_vowels = bytearray(set(range(0x100)) - set(b'aeiou'))" "text.encode('ascii', 'ignore').translate(None, non_vowels).decode('ascii')"
100000 loops, best of 3: 3.06 usec per loop
python3 -m timeit -s "text = 'mississippi'*100;d=dict.fromkeys(i for i in range(127) if chr(i) not in 'aeiou')" "text.translate(d)"
10000 loops, best of 3: 71.3 usec per loop
python3 -m timeit -s "import string; import sys; text='mississippi'*100; m = dict.fromkeys(i for i in range(sys.maxunicode+1) if chr(i) not in 'aeiou')" "text.translate(m)"
10000 loops, best of 3: 71.6 usec per loop
python3 -m timeit -s "text = 'mississippi'*100" "''.join(c for c in text if c in 'aeiou')"
10000 loops, best of 3: 60.1 usec per loop
python3 -m timeit -s "text = 'mississippi'*100" "''.join([c for c in text if c in 'aeiou'])"
10000 loops, best of 3: 53.2 usec per loop
python3 -m timeit -s "import re;text = 'mississippi'*100; p=re.compile(r'[aeiou]')" "''.join(p.findall(text))"
10000 loops, best of 3: 57 usec per loop
The timings in sorted order
translate (bytes)    |  2.88
translate (bytearray)|  3.06
List Comprehension   | 53.2
Regular expressions  | 57.0
Generator exp        | 60.1
dict.fromkeys        | 71.3
translate (unicode)  | 71.6
As you can see the final method using bytes is the fastest. 
python3.5 -m timeit -s "text = 'mississippi'*100; non_vowels = bytes(set(range(0x100)) - set(b'aeiou'))" "text.encode('ascii', 'ignore').translate(None, non_vowels).decode('ascii')"
100000 loops, best of 3: 4.17 usec per loop
python3.5 -m timeit -s "text = 'mississippi'*100; non_vowels = bytearray(set(range(0x100)) - set(b'aeiou'))" "text.encode('ascii', 'ignore').translate(None, non_vowels).decode('ascii')"
100000 loops, best of 3: 4.21 usec per loop
python3.5 -m timeit -s "text = 'mississippi'*100;d=dict.fromkeys(i for i in range(127) if chr(i) not in 'aeiou')" "text.translate(d)"
100000 loops, best of 3: 2.39 usec per loop
python3.5 -m timeit -s "import string; import sys; text='mississippi'*100; m = dict.fromkeys(i for i in range(sys.maxunicode+1) if chr(i) not in 'aeiou')" "text.translate(m)"
100000 loops, best of 3: 2.33 usec per loop
python3.5 -m timeit -s "text = 'mississippi'*100" "''.join(c for c in text if c in 'aeiou')"
10000 loops, best of 3: 97.1 usec per loop
python3.5 -m timeit -s "text = 'mississippi'*100" "''.join([c for c in text if c in 'aeiou'])"
10000 loops, best of 3: 86.6 usec per loop
python3.5 -m timeit -s "import re;text = 'mississippi'*100; p=re.compile(r'[aeiou]')" "''.join(p.findall(text))"
10000 loops, best of 3: 74.3 usec per loop
The timings in sorted order
translate (unicode)  |  2.33
dict.fromkeys        |  2.39
translate (bytes)    |  4.17
translate (bytearray)|  4.21
List Comprehension   | 86.6
Regular expressions  | 74.3
Generator exp        | 97.1
                        You can try pythonic way like this,
In [1]: s = 'mississippi'
In [3]: [char for char in s if char in 'aeiou']
Out[3]: ['i', 'i', 'i', 'i']
Function;
In [4]: def eliminate_consonants(x):
   ...:     return ''.join(char for char in x if char in 'aeiou')
   ...: 
In [5]: print(eliminate_consonants('mississippi'))
iiii
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