Wine is a _translation layer_ (a program loader) capable of running Windows applications on Linux and other POSIX compatible operating systems. Windows programs running in Wine act as native programs would, running without the performance or memory usage penalties of an emulator, with a similar look and feel to other applications on your desktop.
Mono is a _software platform_ designed to allow _developers_ to easily create cross platform applications. It is an open source implementation of Microsoft's .Net Framework based on the ECMA standards for C# and the Common Language Runtime. We feel that by embracing a successful, standardized software platform, we can lower the barriers to producing great applications for Linux.