Understanding journey planner jargon

Berlin
The Deutsche Bahn site lists lots of Berlin stations, but the main one you'll want is 'Berlin Hbf', Berlin main station. Another one you might want if you're flying into or out of what is now the city's only airport is 'Flughafen BER'. But NOT 'Schönefeld'! That is the stop for the no longer used old terminal building. The airport is also listed as 'BER Berlin Brandenburg Airport'.

Cologne/Köln
Cologne and Köln are one and the same place. Cologne is just our anglicised spelling of the German 'Köln'. The Deutsche Bahn website also, for some reason, refers to the city by its old Roman name Colonia. So, for trains to/from Cologne, you want Cologne, Köln or Colonia. If you enter Köln into the DB journey planner, you'll see numerous stations listed. Only 4 are likely to be of any interest to you:

Köln
Enter this and you'll be shown connections to/from Cologne main station (Köln Hbf) and also to/from Köln Messe/Deutz, a large secondary station just over the Rhine from the cathedral and main station.

Köln Hbf
Cologne main station. Right next to the cathedral (Dom) on the banks of the Rhine.

Köln Messe/Deutz
A major secondary station just across the river. In the district of Deutz and adjacent to the trade fair grounds (Messe), hence its name. Easy transport links across the river to the main station, cathedral and old town. Direct transport link to the stadium.

Köln/Bonn Flughafen
Cologne/Bonn Airport, 7 miles south of Cologne. Direct rail link into the city.

Note: If you come across 'Cologne-Deutz foires' that is simply the French name for 'Köln Messe/Deutz'.

Dortmund
The station you want is Dortmund Main Station, i.e. 'Dortmund Hbf'.

Düsseldorf
If you spell the city's name as 'Dusseldorf', the DB site will offer you 'Dusseldorf City'. If you spell it 'Duesseldorf', it will offer you 'Düsseldorf Hbf'. Both then give you train times to/from the 'Main Station' (Hauptbahnhof). If you see 'Düsseldorf Flughafen' or 'Dusseldorf Airport' that's, well, the airport!

Frankfurt
Be aware that there are two places called Frankfurt in Germany! The one where Eintracht play is Frankfurt am Main. The other one is Frankfurt an der Oder, over 380 miles away, on the Polish border. So, when you're looking for trains to Frankfurt to watch Eintracht, you do NOT want 'Frankfurt(Oder)'!!!

Frankfurt(Main) has several railway stations. Frankfurt International Airport also has a mainline railway station. If you're after trains to/from the city's main station, you want 'Frankfurt(Main)Hbf'. If you want the airport station, you need 'Frankfurt(M)Flughafen' or 'Frankfurt(Main) Flughafen Fernbf'. Those mean 'Frankfurt am Main Airport' and 'Frankfurt am Main Airport Mainline Station' respectively. Some long-distance trains also go to/from 'Frankfurt(Main)Süd', Frankfurt am Main South. Like the main station (Hbf), it's pretty central too, just a bit south of the river.

Gelsenkirchen
The station you want is Gelsenkirchen Main Station, i.e. 'Gelsenkirchen Hbf'.

Hamburg
The main station is, of course, 'Hamburg Hbf'. You will also find that some long-distance trains go to/from 'Hamburg-Altona'. It is located some way to the west of the main station and thus closer to the HSV stadium. For the airport, the Deutsche Bahn search facility recognises both 'Flughafen Hamburg' and 'Hamburg Airport'.

Leipzig
The station you want is Leipzig Main Station, i.e. 'Leipzig Hbf'. For the airport you want 'Leipzig/Halle Flughafen' or 'LEJ LeipzigHalleAirp'.

Munich
You surely know that Munich and München are one and the same place, right? So, for Munich main station, the DB site may show you 'Munich (München)' or 'München Hbf'. For the airport it may show you 'München Flughafen Terminal' or 'Munich Airport T'.

Stuttgart
Nothing complicated here. For the main station you want 'Stuttgart Hbf' and for the airport 'Stuttgart Flughafen/Messe'.