NOTE: This reply is a bit old. Since it talks about internals and unstable features, things have changed a little bit. The basic mechanism remains the same though, so the answer is still capable of explaining the underlying mechanisms of box.
What does box x usually uses to allocate and free memory?
The answer is the functions marked with lang items exchange_malloc for allocation and exchange_free for freeing. You can see the implementation of those in the default standard library at heap.rs#L112 and heap.rs#L125.
In the end the box x syntax depends on the following lang items:
owned_box on a Box struct to encapsulate the allocated pointer. This struct does not need a Drop implementation, it is implemented automatically by the compiler. 
exchange_malloc to allocate the memory. 
exchange_free to free the previously allocated memory. 
This can be effectively seen in the lang items chapter of the unstable rust book using this no_std example:
#![feature(lang_items, box_syntax, start, no_std, libc)]
#![no_std]
extern crate libc;
extern {
    fn abort() -> !;
}
#[lang = "owned_box"]
pub struct Box<T>(*mut T);
#[lang = "exchange_malloc"]
unsafe fn allocate(size: usize, _align: usize) -> *mut u8 {
    let p = libc::malloc(size as libc::size_t) as *mut u8;
    // malloc failed
    if p as usize == 0 {
        abort();
    }
    p
}
#[lang = "exchange_free"]
unsafe fn deallocate(ptr: *mut u8, _size: usize, _align: usize) {
    libc::free(ptr as *mut libc::c_void)
}
#[start]
fn main(argc: isize, argv: *const *const u8) -> isize {
    let x = box 1;
    0
}
#[lang = "stack_exhausted"] extern fn stack_exhausted() {}
#[lang = "eh_personality"] extern fn eh_personality() {}
#[lang = "panic_fmt"] fn panic_fmt() -> ! { loop {} }
Notice how Drop was not implemented for the Box struct? Well let's see the LLVM IR generated for main:
define internal i64 @_ZN4main20hbd13b522fdb5b7d4ebaE(i64, i8**) unnamed_addr #1 {
entry-block:
  %argc = alloca i64
  %argv = alloca i8**
  %x = alloca i32*
  store i64 %0, i64* %argc, align 8
  store i8** %1, i8*** %argv, align 8
  %2 = call i8* @_ZN8allocate20hf9df30890c435d76naaE(i64 4, i64 4)
  %3 = bitcast i8* %2 to i32*
  store i32 1, i32* %3, align 4
  store i32* %3, i32** %x, align 8
  call void @"_ZN14Box$LT$i32$GT$9drop.103617h8817b938807fc41eE"(i32** %x)
  ret i64 0
}
The allocate (_ZN8allocate20hf9df30890c435d76naaE) was called as expected to build the Box, meanwhile... Look! A Drop method for the Box (_ZN14Box$LT$i32$GT$9drop.103617h8817b938807fc41eE)! Let's see the IR for this method:
define internal void @"_ZN14Box$LT$i32$GT$9drop.103617h8817b938807fc41eE"(i32**) unnamed_addr #0 {
entry-block:
  %1 = load i32** %0
  %2 = ptrtoint i32* %1 to i64
  %3 = icmp ne i64 %2, 2097865012304223517
  br i1 %3, label %cond, label %next
next:                                             ; preds = %cond, %entry-    block
  ret void
cond:                                             ; preds = %entry-block
  %4 = bitcast i32* %1 to i8*
  call void @_ZN10deallocate20he2bff5e01707ad50VaaE(i8* %4, i64 4, i64 4)
  br label %next
}
There it is, deallocate (ZN10deallocate20he2bff5e01707ad50VaaE) being called on the compiler generated Drop!
Notice even on the standard library the Drop trait is not implemented by user-code. Indeed Box is a bit of a magical struct.