Minecraft is quite CPU intensive. A computer that hosts a Minecraft server should be powerful enough for handling the load. There can be both single as well as multi-player servers. Then again, the system configuration determines how many players can play at the same time.
While the official system requirements for Minecraft Server have not yet been announced, the following is what has been derived from practical set-ups.

System requirements for single player:-

Most present-day computers can easily allow single-player Minecraft gaming. Some of the older ones can do too. Having a 1GB RAM or even 500MB, a single core 1GHz processor, upload and download speeds of 512kbps and 100MB of free disk space is quite sufficient.

System requirements for multi-players:-

When it comes to hosting a Minecraft server for multiple players, it is essential that you have some knowledge about networking and hosting. The upload speed supported by your system and internet connection, plays a major role in determining the number of players that can play at a time. The higher the upload speed, the more the number of people that can play simultaneously. The hard drives that ship with computers these days are spacious enough for handling multiple players. However, if you wish to beef it up, you can always go ahead with it. A dual core processor, with about 1.5GHz speed should be fine. However, if the speed is 2GHz or more, it should be better. The video card you use on the server does not make much of a difference, as graphics is rendered on the system on which the game is played. Many people prefer using the powerful AMD graphics processors and the newer NVidia cards. Even Intel GPUs are used on the gaming machines.
Here is a table which gives a rough idea of the minimum system requirements for Minecraft server hosting for multiple players. Generally, players are limited to 25-30 per server. However, there are a few extremely powerful servers that host up to 100+ players at one time. Such servers are however very expensive to set-up and maintain.

Players	RAM	Upload Speed	Download Speed
2	500 MB or more	512 kbps	1 Mbps or more
3	500 MB or more	1 Mbps	2 Mbps or more
4	500 MB	2 Mbps	3 Mbps or more
6	1 GB	2 Mbps	3 Mbps or more
8	1 GB	3 Mbps	10 Mbps
9	2 GB	3 Mbps	10 Mbps
12	2 GB	4 Mbps	10 Mbps
15	2 GB	5 Mbps or more	10 Mbps
20	3 GB	10 Mbps	10 Mbps
	30	4 GB	10 Mbps	10 Mbps