If you use third party software inside your own programs, and it is licensed (which it most likely will be if it is available to the general public) the requirements for use will be depend on the terms of the license. This will differ from one work to another. For example most proprietary software will not require you to do much of anything, providing that you paid for that license beforehand.
Opensource licenses are different. There are a few different types of license available. Different versions may have different requirements from users. Three things, however, will always be required. First of all, you need to have a copy of the license. Just like a driver's license is proof that you are allowed to drive, a software user license proves that you are allowed to use that software.
The first of these requirements sort of implies the second. That is, you are required to be bound by and/or to carry out any terms to which you agree on that license. This is where the "mention the author in your work thing comes in." Note that this requirement isn't technically a requirement of open source licensing. It isn't written somewhere in the ten commandments of opensource (or in any law) that you must do anything printed on a license to use the app. However, remember, having a valid license is required by law. Therefore, Developers only need to state "this license is only valid if these demands are met." It isn't that much different than proprietary. Paid purchase is more or less just another demand prerequisite to license validation.
Now the third rule (requirement) is not only important, it is the very heart of open-source. It states that you must include the open-source. Believe it or not, this is where open-source gets it's name! (and water is indeed wet.)
This just means that, no matter how you decide to distribute your application, be it via package manager, as a clickable download from a website, on CD-ROM via snail mail, or by any other method, the source code has to be somehow publicly available. If you distribute a java app as a jar, you need to either include a copy of the source code, provide a link to it, or at least provide directions for how to obtain it. Note that in the case of included third-party software, this was already the responsibility of the original developers. Therefore, it should have already been included (or pointed to) as part of the software package that you received. However, since you are now using that code, and since it is still open-source, the requirement falls on you to make sure that it stays in tact. Just ask Android.
There are a few other requirements, laid out by the Open-Source Initiative, but at the end of the day, they are just a standards group. When I refer to open-source, I am speaking of the free and often community-built, but licensed software.