If your spouse is a non-resident alien, you will generally file using "married filing separately." You can't e-file without the SSN or ITIN, so print your return and mail it in and write "NRA" in the space for their tax number.
If your spouse is a resident alien, you can file separately or jointly. If separately, print your return and write in "NRA". If filing jointly, print your return, leave the space for their SSN blank, and attach a W-7 form to your return to request an ITIN, and mail them in together.
If your spouse is a non-resident alien, you have the option to declare them a resident alien for tax purposes. This means you can file jointly, which has lower tax rates and better deductions than married filing separately. But it also means you must report and pay tax on your spouse's world-wide income (subject to a credit for foreign taxes paid.) If your spouse has low or no income this may save you money, but if they have high income it may cost you more in taxes. See the links below for more information.
https://www.irs.gov/individuals/international-taxpayers/taxation-of-nonresident-aliens
https://www.irs.gov/individuals/international-taxpayers/determining-alien-tax-status
https://www.irs.gov/pub/irs-pdf/fw7.pdf
