The Apache License is a permissive license. It basically lets you do pretty much anything you want, as long as you tell people you used the software, and that it was Apache Licensed.
[I]f part of my application uses code I did not write ... under the Apache 2.0 license, am I not allowed to charge for my app?
No, it is perfectly ok to charge for your app. The Apache license does not prevent you from charging for applications that use Apache Licensed code.
From the license:
You may reproduce and distribute copies of the Work or Derivative Works thereof in any medium, with or without modifications, and in Source or Object form, provided that You meet the following conditions:
There is no mention in those conditions about charging or not charging money.
Do I have to release the source to my entire project?
No. The conditions from the license do not mention any requirement to release source.
What am I allowed to do and not allowed to do?
From the conditions:
You must give any other recipients of the Work or Derivative Works a copy of this License; and
I.e. you must include the license file with your application. (For binary applications, most people put this in a "terms of service" menu or some such.)
You must cause any modified files to carry prominent notices stating that You changed the files; and
I.e. if you give someone the source, you must point out explicitly what was changed.
You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and
I.e. you cannot delete any license notices from the source package
*If the Work includes a "NOTICE" text file as part of its distribution, then any Derivative Works that You distribute must include a readable copy of the attribution notices contained within such NOTICE file, excluding those notices that do not pertain to any part of the Derivative Works, in at least one of the following places: within a NOTICE text file distributed as part of the Derivative Works; within the Source form or documentation, if provided along with the Derivative Works; or, within a display generated by the Derivative Works, if and wherever such third-party notices normally appear. The contents of the NOTICE file are for informational purposes only and do not modify the License. You may add Your own attribution notices within Derivative Works that You distribute, alongside or as an addendum to the NOTICE text from the Work, provided that such additional attribution notices cannot be construed as modifying the License. *
I.e. if you have your own NOTICE file, you must include all license information from the original.
Note that this applies to the Apache License. There is another very common license, the MIT license, that is functionally equivalent.
The other most common licenses are the GPL and the LGPL. Both of these licenses also let you charge money for applications generated, but different in that they both require you release source to users who you give binaries to, always for the GPL, under specific conditions for the LGPL.