What is a "Magic Packet" for waking a computer?

Sam3000's answer is very nice. I'll add some technical details.

Wake on Magic Packet causes the network card to awaken the computer when it receives a magic packet. A packet is considered "magic" when it contains FF FF FF FF FF FF (six instances of the largest possible byte value) followed by sixteen instances of the card's six-byte MAC address. That sequence can appear anywhere within the frame, so the packet can be sent over any higher-level protocol. Usually, UDP is used, but sometimes raw frames with EtherType 0x0842 are used. (Source: Wikipedia.)

Wake on Pattern Match is a superset of the previous. It will cause the card to wake the machine when various things come in, including a magic packet, a NetBIOS name query, a TCP SYN packet (either TCPv4 or TCPv6), etc. Those last ones may require ARP offload to be enabled. (Source: TechNet.)

If you don't want/need your computer to be woken up from anywhere else, you can disable both of those options.

These two settings form a feature of most modern computers known as "Wake on LAN"; in a nutshell, leaving this setting on allows the network card of your system to receive sufficient power to remain in standby mode, while the rest of the system is powered off. While in standby mode, it may receive a "magic packet" - a small amount of data specific to the mac address of the network card - and will respond to this by turning on the system. Very useful for remote control situations, however you are absolutely right that you may disable these features without any negative consequences - kudos to you for doing some prior research too.

For further information read "How-To Geek Explains: What is Wake-on-LAN and How Do I Enable It?".