JavaScript (Node.js), 23 bytes

f=x=>x?x%2+f(x-1>>1):''

Try it online!

Shorter than some builtins

05AB1E (legacy), 3 bytes

>b¦

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Haskell, 41 38 30 29 bytes

l="":[b:x|x<-l,b<-"01"]
(l!!)

Usage example: (l!!) 4 -> "10".

Starting with the empty list as the first element, walk lazily through the list and append the current element with 0 and with 1 in front of it.

Edit: @xnor saved 3 11 bytes. Thanks!

C, 40 bytes

f(n){if(++n/2)putchar(n%2+48),f(n/2-1);}

Converts the input to bijective base 2 (with symbols 0 and 1), like the other answers.

ideone it!

Haskell, 35 bytes

h 1=[]
h n=mod n 2:h(div n 2)
h.(+1)

Haskell doesn't have a binary built-in, so the (reversed) conversion is done manually. To remove the initial 1, the base case has 1 transform to the empty list.

Edit: Saved a byte by conjugating by the +1 instead.

h 0=[]
h m=1-mod m 2:h(div(m+1)2-1)

Jolf, 4 bytes

Try it here!

wBhj
  hj  input + 1
 B    converted to binary
w     with first removed.

Very simple strategy, also happens to be the shortest.

JavaScript (ES6), 29 bytes

x=>(~x).toString(2).slice(2)

Same idea as xnor.

f=x=>(~x).toString(2).slice(2);
[...Array(100)].map((v,x)=>A.textContent+=x + ': ' + f(x) + '\n')

<pre id=A></pre>

Python, 20 bytes

lambda n:bin(~n)[4:]

Test:

>> [bin(~n)[4:] for n in range(16)]
['', '0', '1', '00', '01', '10', '11', '000', '001', '010', '011', '100', '101', '110', '111', '0000']

Doing lambda n:bin(n+1)[3:] increments the input, then takes the binary representation with the first symbol removed ([3:] because the prefix 0b is two chars chars). Since any positive number starts with 1 in binary, this uniquely gives a binary representation.

A byte is saved by instead using the bit complement ~n to get the negation -(n+1), and removing the negative sign by chopping off one more symbol.

Pyth, 5 bytes

t.BhQ

Simply a translation of xnor's answer into Pyth.

Q is eval()'d input(), h increments it, .B converts it to a binary string, and t takes the "tail" (which is everything except the first character).

Jelly, 3 bytes

‘BḊ

Same idea as xnor: maps 0 1 2 3 4 ... to [] [0] [1] [0 0] [0 1] ...; the code is basically increment → binary → remove first.

Try it out online.