Whether or not you can change your fiscal year end depends on how your business is structured. Sole proprietorships, members of a partnership in which all the partners are individuals and corporations are the only forms of business that can have a fiscal year end other than December 31st.
If you're starting a business, you can change your fiscal year end simply by choosing to have a different fiscal year end. To do this, you need to fill out Form T1139, Reconciliation of Business Income for Tax Purposes and file the form by the required date to file your election.
If your business already have an established fiscal year end date and you want to change it, you need to submit a written request for approval of a change in your fiscal year period to the Director of your local district taxation office.
There are two points about a change in fiscal year end to keep in mind:
1) Changes to your fiscal year period that occur because the fiscal year period is "revised by operation of law" do not need to be approved.
So, for instance, if your business's fiscal year period changes because a sole proprietor dies or sells the business, because a partnership ceases to exist, or because a corporation goes bankrupt, you don't have to request approval for a change in your fiscal year end.
2) The CRA (Canada Revenue Agency) will only approve changes to your fiscal year period that it considers to be made for "sound business reasons".
The CRA's stated purpose for requiring approval of changes to fiscal periods is "to prevent taxpayers from rearranging their fiscal periods primarily to minimize taxes". Their archived tax bulletin IT-179R Change of fiscal period gives examples of sound business reasons such as a business wanting to change its fiscal year end date to coincide with a seasonally slack period or a corporation changing its fiscal period so that its fiscal year end date is the same as its parent company.