Well, in .NET 1.1 and 2.0 there are methods in the System.Net namespace that do that. Speaking of System.Net, don't forget to include the using statement, to avoid the long lines I have below,
using System.Net;
// Get the hostname
string myHost = System.Net.Dns.GetHostName();
// Show the hostname
MessageBox.Show(myHost);
// Get the IP from the host name
string myIP = System.Net.Dns.GetHostByName(myHost).AddressList[0].ToString();
// Show the IP
MessageBox.Show(myIP);