Enable Unknown Sources
Before attempting a manual installation of apps using the .apk files, you must first allow your phone to install from "Unknown Sources" (i.e. non-Market apps).
To do this, navigate to Menu -> Settings -> Applications and check the box marked "Unknown Sources".
1. Installing app Using The Conventional Method
Step 1: Install File Manager
Android does not natively come with any method of browsing the data on your SD card, so you will need to install a file manager from a market. There are a large variety of file managers available on Android, but my personal favourite is ASTRO File Manager.
Step 2: Copy .apk file to SD card
Once you have ASTRO File Manager installed, connect your Android device to your PC using your USB cable. Mount the SD card and copy over the .apk file you would like to install.
Step 3: Install .apk
On your Android device, navigate to the .apk file using ASTRO File Manager and select it.
This will open a dialog box allowing you to install the app. Select "Open App Manager".
On the next two pages, select "Install" and "Install" again to install the .apk.
Your new app is now installed.
2. Installing app using Dropbox
Dropbox is really a versatile app and is a waste if we don't fully utilize it. The method is simple. In your computer, download the apk file to your Dropbox folder. Let it finish syncing. In your phone, open the Dropbox app, navigate to the folder where you keep the apk file, click on it. Dropbox will then download the apk file. The usual installation follows.
3. Installing app Using the Online Apk Installer
The online apk installer is a web app created by a helpful XDA-Developers forum member htc-hd2, with the aim to make manual installation of an .apk file much easier. It is useful if you want to share an apk file with a friend.
Step 1: Upload File
First, navigate to www.apkinstall.com.
Here you will see a black "Browse Files…" button. Select this and choose the apk file on your PC.
The apk file will remain active on the website for 15 minutes.
Step 2: Scan QR Code
Once you have uploaded the file, a QR code will appear on the website.
You must scan this QR code with a Barcode Scanner. The website suggests using Barcode Scanner, however I personally prefer using Google Goggles.
Once the QR code has scanned, you can click on the link to download the .apk file straight to your device.
Step 3: Install .apk
After the .apk file has downloaded to your Android device you can install it by simply clicking on it and navigating through the installation pages.