IP spoofing is filling in the IP address field on a packet with an address that isn't the sender's IP address. This means you can't receive responses to that packet, so it isn't particularly useful, but it can be used as part of an exploit in order to make it harder to trace, or to make it look like the packet came from another source in order to bypass IP based authentication measures (obviously only if the exploit doesn't require a response from the server). Another use is when carrying out DDOS attacks - here you don't care where the response is sent as long as the server handles the request.
An ARP spoofing attack involves transmitting faked ARP packets. The Address Resolution Protocol is used for translating network addresses (IP addresses) into link layer addresses (MAC addresses).
What normally happens is, a computer sends out an ARP request to every device on the network, asking for the MAC address corresponding to a given IP. The device that currently has that IP address then sends an ARP response back to the computer that originally made that request, containing it's MAC address.
In an ARP spoofing attack, you send a fake ARP response with your MAC address, before the intended recipient can respond. Now the computer that made the request thinks that that IP address belongs to you, and it will send all traffic that was intended for that recipient on to you instead.
You can then forward the traffic back on to the original recipient - you don't have to do this, but it allows you to intercept the connection without either device being aware. If the device that you have ARP hijacked is the gateway, you can now intercept (and tamper with) all internet traffic on the network (provided it isn't encrypted