Pip -l, 10 8 bytes

RZJsXg.0

Takes the integers as command-line arguments. Outputs using 0 in place of x. Attempt This Online!

Explanation

RZJsXg.0
          g is list of cmdline args; s is space (implicit)
   sXg    Convert each arg to a string of that many spaces
      .0  Append 0 to each string of spaces
 ZJ       Transpose, padding to a rectangle with spaces
R         Reverse the resulting list
          Autoprint, one row per line (-l flag)

Python 3, 67 bytes

lambda x:[[v==i and'*'or' 'for v in x]for i in range(max(x),-1,-1)]

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for i in range(max(x),-1,-1) -> identify how high the mountain will be, max value is first line (top), so start from max, iterate to 0

[v==i and'*' or' ' for v in x] -> for every integer in input set it as * if height matches current line, set it whitespace otherwise.

[EDIT] as per @hyper-neutrino suggestion to get 3 bytes less, and wrapped in lambda as per @Razetime comment

R, 65 bytes

function(a)for(i in max(a):0)cat(c(' ','x')[1+(a==i)],'
',sep='')

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Pyth, 7 bytes

_.tm+*;

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Canvas, 7 bytes

｛ ×x＋］↶

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Jelly, 6 bytes

=ⱮṀṚo⁶

A monadic Link that accepts a list of positive integers (the 1-indexed option) and yields a list of lists of characters and integers (1 being the choice for x).

Try it online! (The footer joins with newlines and then Jelly's implicit, smashing print produces the output.)

How?

=ⱮṀṚo⁶ - Link: list of integer heights, H
  Ṁ    - maximum (H) -> M
 Ɱ     - map across (h in [1..M]) with:
=      -   (H) equals (h) (vectorises)
   Ṛ   - reverse -> X
     ⁶ - space character
    o  - (X) logical OR (' ') (vectorises)

JavaScript (Node.js), _{^{77 74 73}} 70 bytes

-1 thanks to @RecursiveCo.
-3 thanks to @m90's idea

a=>{for(i=Math.max(...a);i;i--)console.log(a.map(e=>i^e&&' ').join``)}

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Takes one-indexed input.

J, 17 bytes

[:|.@|:' x'#~,.&1

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^{+6 thanks to Lynn for spotting a bug in the original solution: ' x'{~=\:~. This approach can be made to work but it's no longer golfy: ' x'{~]=/~[:i.1-@+>./}

how

• ,.&1 Zip input with 1:

  1 1
  0 1
  0 1
  1 1
  2 1
  4 1
  2 1
  0 1
  

• ' x'#~ Duplicate ' x' according to these pairs

   x   
  x    
  x    
   x   
    x  
      x
    x  
  x
  

• [:|.@|: Rotate left:

       x  
  
      x x 
  x  x    
   xx    x
  

Octave / MATLAB, 43 bytes

@(x)flip([accumarray([x find(x)],3)+32 ''])

Anonymous function that inputs a 1-based column vector and outputs a char matrix with # and

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Vyxal, 11 9 bytes

-2 bytes through auto-vectorizing

ð*×+ðÞṪṘ⁋

ð*            # vectorised n spaces
  ×+          # append "*"
     ðÞṪ      # transpose with space as filler
        Ṙ⁋    # reverse and join by newline

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Wolfram Language (Mathematica), 44 bytes

Print@@@Table[" "Unitize[#+i],{i,-Max@#,0}]&

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Prints the mountain, using 0 instead of x.

Remove the Print@@@ for -8 bytes if an array of characters and numbers is acceptable output.

Zsh, ⁶⁶ 63 bytes

s=${(l/${${(On)@}[1]}*#/)}
for x;s[++i+$#s-#*x]=.
fold -$#<<<$s

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-3 thanks to JoKing, but still probably too long.

Explanation:

• ${${(On)@}[1]}: get the maximum input
• s=${(l/*#/)}: construct a string of spaces which is (number of inputs) × (maximum of inputs) in length
• for x;s[++i+$#s-#*x]=.: set the correct index of the string to a . for each element
• fold -$#<<<$s: line-wrap the string at (number of inputs) in length

Ruby, 51 bytes

->a{(-a.max..0).map{|y|a.map{|x|x==-y ??x:' '}*''}}

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Jelly, 7 bytes

Ṭ€z0Ṛo⁶

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Outputs a list of lines. +1 byte (append Y) if unacceptable.

Takes input 1 indexed, uses 1 as the mountain character

How it works

Ṭ€z0Ṛo⁶ - Main link. Takes a list L on the left
 €      - Over each element in L:
Ṭ       -   Untruthy; Map each n to [0, 0, ..., 1] with n-1 zeros
  z0    - Transpose, padding with zeros
    Ṛ   - Reverse
     o⁶ - Replace 0s with spaces

Red, 91 bytes

func[a][i: last sort copy a until[forall a[prin either a/1 = i[1][sp]]print""0 > i: i - 1]]

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-6 bytes from Galen Ivanov.

Japt -R, 7 bytes

1-indexed, using " for the peaks.

£QùXÃÕÔ

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£QùXÃÕÔ     :Implicit input of array
£           :Map each X
 Q          :  Quotation mark
  ùX        :  Left pad with spaces to length X
    Ã       :End map
     Õ      :Transpose
      Ô     :Reverse
            :Implicit output joined with newlines

05AB1E (legacy), 5 bytes

$úζR»

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$        # push "1" and the input
 ú       # for each integer in the input,
         # pad "1" with this many spaces in the front
  ζ      # tranpose, padding with spaces
   R     # reverse, the mountains should not be upside down
    »    # join by newlines

Vyxal, 4 bytes

×꘍R§

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That's right, it's flagless.

Explained

×꘍    # [(n * " ") + "*" for n in input]
  R   # [x[::-1] for x in ^]
   §  # vertically join (rotate) ^

Using the L flag would make it 3 bytes.

Charcoal, 6 bytes

↑ＥＡ◧xι

Try it online! Link is to verbose version of code. 1-indexed. Explanation:

  Ａ     Input array
 Ｅ      Map over elements
    x   Literal string `x`
   ◧    Left-padded to width
     ι  Current element
↑       Print rotated

APL(Dyalog Unicode), 18 bytes ^SBCS

{⊖⍉⍕⍪'x',¨⍨⍵⍴¨' '}

{          ⍝ Open function
⊖⍉         ⍝ Flip diagonally and then vertically
⍕          ⍝ Make table into multiline string
⍪           ⍝ Display one element per row
'x',¨⍨      ⍝ Append an 'x' to each element
⍵⍴¨' '      ⍝ Duplicate each space 'input' times
}           ⍝ Close function

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Python 2, 66 bytes

a=input()
n=max(a)
while~n:print''.join(' x'[x==n]for x in a);n-=1

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