You can make advance reservations for seats on major European Trains through various travel agents. For contact details, visit this website.
At the European train station
At larger train stations in Europe, like the ones with a Eurail Aid Office, you can make reservations at a ticket window. Just show your Eurail Pass and tell the official which train(s) you want to reserve.
Through a Railway by phone / email
Call centres of European Railways can offer most trains, with some exceptions, like most French trains. Some suggested call centres:
Austrian Railways ÖBB have a CallCentre which handles reservations with English speaking staff.
German Railways UK Booking Centre has English speaking staff. They can book seat, sleeper or couchette reservations throughout Europe. You can contact the Booking Centre through this booking form (mention that you have a Eurail Pass) or by phone: (+44) 08718 80 80 66 (Monday to Friday 9am to 8pm, Saturday and Sunday 9am to 1pm CET).
When booking, be sure to mention that you travel with a Eurail Pass and that you only need a reservation. Booking fees may apply.
Through a Railway website
There isn't a comprehensive Railway website where you can make online reservations for all European trains. The online reservation options are increasing though, especially for domestic trains. Below you can find instructions for some countries: