Any member of parliament can move a bill.
A member of the public can't, for example, mail a bill in to the Speaker's office and say 'here you go, get them to vote on that.' A member of one of the houses of parliament has to put it on the notice paper.
Normally it is the Government of the day that introduces bills. Ordinarily it would be a Minister who would do this after receiving approval from Cabinet.
When someone other than a Minister introduces a bill, it is called a 'private member's bill'.
A Minister would not introduce a bill without Cabinet approval (that would make the Government look bad). A non-Minister Government MP (backbencher) may introduce a private member's bill and this could be a genuinely independent effort or it could be a way for the Government to distance itself from something controversial.
The Government may control the business of the chamber in question (particularly if that is the House of Commons), and may be able to defer consideration of a private member's bill indefinitely if they really want to.
See also Wikipedia.
In terms of drafting, there are public servants who draft Government bills. It is up to the Government to decide whether to allow other members of parliament to have access to these resources. A private member's bill may be drafted by the member personally, by a lobbyist or by anyone with a word processor.