There are a lot of free apps from the Google Play (previously known as Android Market) where you can download and install it on your Android device. When you want to install an app, all you need to do is click on the DOWNLOAD button and it will start downloading an APK file and automatically install. Normally there is no need to save the APK file after installation but if you need it, one painful way is to download the APK file to your android phone, use Astro to backup it to SDCard, connect phone to PC, then copy the APK file to computer.
redphx from Vietnam who is an android app developer has made all of that easier by creating an extension called APK Downloader that allows you to easily download APK files from Google Play with one click using Google Chrome web browser.
Enter the email and password that you use on the Google Play (Android Market), and to obtain your Android Device ID, you can dial *#*#8255#*#* which will open “GTalk Service Monitor”and look for the lines that begins with JID and Device ID. Your email is JID, and your device id is a string that after android- prefix. Once you’re logged in, go to Google Play and select any free apps that you want to download to your computer. You will notice that the address bar contains an APK Downloader icon and clicking on it will open a new tab allowing you to start downloading the APK file.
Do note that APK Downloader only works on Google Play but not on Chrome Web Store.