Sometimes called The Bonn Republic these days by historians, the Federal Republic of Germany (Bundesrepublik Deutschland) was formed out of three of the occupation zones of Germany, the other becoming East Germany.
West Germany became a democracy and an economic superpower. It joined NATO and had a considerable number of foreign military bases there. The bridges had speed limits for tanks (and still do in The Berlin Republic), because World War III would have been fought there.
West Germany had the legacy of the Second World War to deal with. They implemented a No Swastikas policy and banned all vaguely Nazi organisations. West Germany also became a pretty peaceful nation, not getting involved in any foreign wars. This didn't stop East Germany from calling them fascists on a regular basis; the Berlin Wall was called "der antifaschistische Schutzwall," literally meaning "the Anti-Fascist Protection Barrier."
It considered itself a reconstituted Weimar Republic (despite the irony of Weimar now being in East Germany), deciding to pay reparations for both wars and keeping the D license plate. It also elected to pay reparations to Israel starting in 1953 as penance for The Holocaust (and on a more practical level, payment for the labor and property stolen from Jews by the Nazis); Germany remains Israel's foremost advocate in Europe to this day, and has discovered rather interesting ways to show its sorriness to Israel (including giving the IDF nuclear-capable submarines). Contrary to popular belief the capital of West Germany was always Berlin, but given the political reality the national government was located in Bonn, A Small Town in Germany. West Berlin, an occupied city, sent non-voting representatives to the German Parliament.
West Germany also developed culturally, with West German literature, philosophy, films, and music making their mark. On the high end, the likes of Gruppe '47 (including, most famously, Günter Grass) pointed the West German reaction to the horrors of World War II (in essence, "we were guilty and foolish, all of us!") and the process of denazification (they were for it). On the pop-cultural level, The Beatles and countless other British bands had their first breaks in Germany--typically Hamburg--and the West Germans were the first Continental Europeans to really get what this "rock" thing was all about. The Germans returned the favor: late '60s and early '70s German rock (called "Kraut Rock" by the British music press), led by bands like Neu!, Amon Düül II, and early Kraftwerk, had a powerful impact on Post Punk, New Wave, Electronic, and Industrial music. West Berlin in particular was famous for both its rollicking club scene and its Hansa-By-The-Wall (yes, that Wall) recording studio, which was a magnet for musicians German and non-German alike. David Bowie spent most of his most productive and creative period in Berlin (termed, fittingly, his "Berlin period"), inspired by the German scene. Iggy Pop was similarly inspired, recording part of his debut album and all of Lust for Life (you know, the famous one) at Hansa-By-The-Wall.
On another cultural note, the West Germans also managed to create a brilliant national soccer team, winning The World Cup in 1954, 1974, and 1990 (just before reunification). The win in 1954, against Hungary, was a massive boost to West German pride (which until then had been rather shaky), and was seen as a moral victory for the West over the Soviet bloc.
Older sources will sometimes refer to this place as simply "Germany", possibly due to the feeling that this was the real Germany- the other one was just Commie Land with Germans. Bonn itself felt that for a while, refusing to recognise any country bar the USSR that had any relations with the GDR until Willy Brandt's Neue Ostpolitik of the 1970s. The two Germanies recognised each other (but not completely: for example, no embassies, but permanent representatives [Ständige Vertretung] – this would become important in 1990) and joined the United Nations together.
In the early 1980s, it was a site for the US Cruise and Pershing II deployments, something that caused considerable anxiety in a country that would have had nukes from both sides land on it in a nuclear war. In 1986, Chancellor Helmut Kohl announced the unilateral removal of those missiles- a year later the entire lot were got rid of under the INF Treaty.
The German Basic Law was aimed at the reunification of Germany. When Germany reunified, East Germany was merely absorbed into West Germany, creating the reunified Germany- The Berlin Republic. The Basic Law stayed in place with minor changes as it had served the country well.
Media set in West Germany:
- A Small Town In Germany
- The ODESSA File
- The Baader Meinhof Complex
- Recurring Saturday Night Live segment Sprockets
- A number of German Media most non-Germans never ever heard of
- This policy of standing by Israel has occasionally bitten Germany in the arse, as when it threatened to ruin anything resembling a unified European front on the Palestinian statehood resolution in 2011.