Most travellers think "European trains" as a single network. It's actually a patchwork of national rail systems with a handful of named cross-border services bridging them. Knowing which is which saves you from buying tickets you don't need.
Within national borders, every major Western country has its own high-speed brand: TGV inOui in France, ICE in Germany, AVE in Spain, Frecciarossa in Italy, AVE in Spain, ÖBB Railjet in Austria. These are reliable, frequent, and run at 250-320 km/h between most major-city pairs of their country.
Cross-border is where it gets messy. The named services worth knowing:
Eurostar (Brussels, Paris, Amsterdam, Cologne, London via the Channel Tunnel). This is the only train that actually crosses the Channel, and the only practical alternative to flying between London and the continent. Tickets bought 2-3 months ahead can be cheaper than budget airlines.
Railjet (Vienna axis — Munich, Zurich, Budapest, Prague, Venice). Ten-car ÖBB trains running at 230 km/h with a panoramic restaurant car. Often the most pleasant way to cross Central Europe.
TGV Lyria (Paris–Switzerland). Brings TGV technology to Zurich, Geneva, Lausanne and Basel.
EuroCity (a brand label, not one operator). EuroCity trains run scheduled cross-border routes across most of continental Europe — Berlin to Warsaw, Hamburg to Copenhagen, Munich to Italy. They're typically slower than high-speed but reliable.
The gaps. The Balkans (Croatia, Serbia, Bulgaria, Romania) have functioning national networks but very few cross-border high-speed services. Greece is connected only via slower regional links. The Baltics (Estonia, Latvia, Lithuania) are not yet rail-connected to Poland for through services, though the Rail Baltica project is gradually changing that.
Iceland and Ireland have no high-speed rail and are not connected to the continental rail network at all — flying is the only option crossing the water.
If you're planning a multi-country trip, the question is rarely "is there a train?" — it's "which corridor does my route fall on?" Cross-border between Germany and France or France and Italy: high-speed. Between Croatia and Romania: bus is faster.