If you and your spouse is filing jointly - you only file one extension - enter your name and your spouse name.
If you and your spouse is filing separately - file both of you should file extension separately
Depending upon your filing status of prior and current year filing you may be able to use one form 4868 for the extension of both marital members (spouses). The option for the IRS is whether or not you file "jointly" or not. The previous answer is correct with the dependencies.
So, if you complete form 4868 MANUALLY, you have the choice to place your partners/spouses SSN in block 3 to assure Married Filing Jointly filing is extended for both. If you file separately it is just as simple except you leave off the spouses SSN from block 3.
I typically use TurboTax (especially this year for 2018 filing) and the filed extension software left off my spouses SSN from block 3, causing me to file a separate F4868 manually for my spouse.
Be warry of technology solutions that DO NOT work properly because you could find one of you gets an extension and the other could pay a hefty penalty for tax evasion. Seriously, that simple of a mistake by the SW could cause you huge problems.