I realize there's already an answer, but I'd like to go into more detail.
There are generally two kinds of open source software licenses: permissive and copyleft (the latter sometimes called viral).
Both are intended to allow people to freely use, modify, and redistribute a work while ensuring the original author gets credit. This contrasts with releasing your work into the public domain, which would allow anyone do literally anything, even claim to be the author.
The difference is that permissive licenses allow you to re-license your derivative works however you like, while copyleft licenses require your derivatives to be licensed under the same license (hence the term viral). The reasons you'd choose either one are philosophical and political and beyond the scope of this answer.
Both the Apache License and the MIT license are permissive, so incorporating MIT licensed code into your Apache licensed project is certainly allowed. Just be sure to attribute the original author for the parts your incorporated and include a copy of the MIT License terms, as required by the license. Go ahead and read it to make sure you get all the details right; it's fairly short and easy for a layperson to comprehend.