NAS/Devices

Requirements

 * Reliability - The server has to remain online 24/7. If you're far away from your server, you're not going to be able to make physical repairs, so the board had better be reliable.
 * High-speed Storage Options - The main purpose of a server is to host files, so ideally, the hard drive should connect using SATA or USB 3.0.
 * Low Power Usage - It's just a home server, so you don't want to spend too much on electricity.

Uses

 * VPN - A sort of upgraded, encrypted form of a proxy. Great for bypassing paywalls when you visit other countries, or just keeping your internet browsing and cookies safe from snoops.
 * Media Streaming Server - Stream your videos, TV Shows, and movies anywhere in the world, or just anywhere in the house.
 * Personal Cloud Storage (File Server) - If you're away from the house often, want terabytes of cloud storage on the cheap, or just don't trust cloud providers; build your own personal cloud file server, so you can access any of your Terabytes of information anywhere in the world. No third-parties or service charges.
 * Torrent Seedbox - Don't leave your laptop on all day. Have a server do the mundane work of torrenting for you. Also makes it possible to permanently seed torrents on the cheap.

Raspberry Pi
A $35 ARM board. Uses less power than a lightbulb. Unfortunately, it only uses a USB 2.0 bus, which can slow access speeds down quite a bit (especially torrents), so basic, low-traffic server tasks only.

ODROID
If you're looking for something more powerful than the Raspberry Pi, but under $100, the ODROID is for you. It uses the same Exynos processors as the famed Samsung Galaxy S series.

BeagleBone Black
The BeagleBone Black is one of a few fully open source single board computers in the market. If you need a guranteeably secure SBC, this is the way to go. With an ARM Cortex-A8, it's computing power is comparable to one of the Raspberry Pi.

BeagleBone_Black/OpenBSD_Serial_Install

Intel NUC
The Intel Next Unit of Computing is the Cadillac of sub-$200 server boxes. If you need a powerful x86 home server with SATA support, this is the way to go.

Tutorial