The first thing you will need is the actual SDK Manager. You can download the command line version from the official website, but I found a link to the Windows GUI version via a helpful Stack Overflow question. The official link should provide the GUI version for both Mac OS and Linux. As of this answer, this download is 145Mb.
Note that if you intend to use Android Studio at some point, you should just download Android Studio. This will come with the SDK Manager, though it will greatly increase the size of your download. Android Studio is currently quoted at 1, 674Mb, which is grossly oversized, in comparison to the other files and the simple fact that it is not a requirement.
If you do not have it, you will also need to install the Java Development Kit. The current version is about 200Mb1, and you will need to have it installed for the SDK Manager to complete it's installation.
From here, your actually on easy street. You should only need the latest SDK Platform. As of this answer, that version is API 25, also known as Android Nougat 7.1.1. This file is listed as exactly 81.5Mb.
Since you are already in the SDK Manager, it would help to also download the Android USB Driver. You will find this under "Extras", at a meager 8.3Mb. These drivers are often required in order to play test your games, using the Unity Remote app.
Do I need to download the latest version?
Yes and no. With each API release, Google have added more features and change to how the API works. If your project only used features up to API Level 15, for example, you would only need to install API Level 15. It is far easier to simply keep up to date than to ensure you have the API up to the level you require. According to Unity documentation, the bare minimum required is API Level 9, also known as Android Gingerbread 2.3.
What if my players have an older version of Android?
Older versions of android should still support the language of newer APIs - they just won't necessarily support the newer features. This is something you want to take into account; if only to ensure that you build your game to the correct minimum API level. You may also want to consider the percentage of users using each API level.
Do I need to download the additional images?
No. The additional images are merely device images. If you were to be using Android Studio, and you wanted to run your application on a particular hardware emulator, that is when you would need to have the appropriate images installed.
I am making a game for the phone - don't I need the phone images?
No. Assuming you have an android phone, you should use your physical phone to test your game. The image is only for running emulated Android on your computer; to my knowledge, it still requires being run from Android Studio, and it is known to be a bit unpredictable in regards to performance.
All in all, what are the damages?
Assuming the bare necessities, including the USB drivers, you need the following:
- Android SDK Manager: 145Mb
- Java Development Kit: 200Mb
- Latest SDK Platform: 81.5Mb
- Android USB Drivers: 8.3Mb
Assuming you have already installed the Android SDK Manager, you only have a meager 89.8Mb of download left to complete. Assuming you start from scratch, your looking at a download of 434.8Mb. This is still less than 30% of the download required to use Android Studio, the official Android development software.
1 Linux and Mac versions are considerably smaller.