Hi,
I am a big fan of strategy games especially if they are turn based. Then I can take my own time, think and decide what I am doing next. Real time with a pause function that allows giving orders during the pauses is also ok.
Now I ask if you know any good strategy, preferably DRM free, open source or with a disc check. They can be commercial, old or new, but should not require online activation and should be available somehow. The graphics and sound quality is not really important, but the gameplay should be catchy.
Good examples are Master of Orion 1-2, Civilization 1-4, Colonization, Alpha Centaury, Panzer General, X-Com, Hearts of Iron, Battle for Wesnoth...
The number of good turn based strategy games in the last years has been really small. But maybe there are others out there which I am not aware of, modern ones, that you know and like? Please give me a hint about them!

Disciples Series, King's Bounty Series, Worms maybe it is turn based too, Europa Universalis, Victoria 2 and 1, Hearts of Iron Series, Elven Legacy, Total War Series also have good turn based strategy section.

King's Bounty is real time while exploring the map but fights are turn based.

Total War Series has both turn based strategic map sections and real time tactical battles with a pause. Best of both worlds. :D

Elemental comes out tomorrow..

Have you tried Armageddon Empires? I played it to death when i first got it and i still go back regularly so i can curse loudly at 'unfair' dice rolls and my poor opening cards. Great game though.

You made the example of X-Com, but not Jagged Alliance 1 or 2. While JA is mostly "real time" when not in combat, but when there's fighting involved, you switch to turn based mode. You can get reviews on both first and second instalment here on GOG.
Or you could try out Silent Storm. It's real time/turn based Jagged Alliance-ish ww2 themed tactical squad control game where you must infiltrate the third reich (or if you play the germans (!), infiltrate the allied territories), and while dispatching your enemies you must succeed in your mission objectives. Like in JA, combat is turn based while fighting, and real time when not fighting.
I can also think of Massive ASSault, didn't play it much, but you build units that basically have the same characteristics as in PG series.
My mind is drawing blank right now, if I remember any more turn based games, I'll let you know.

Silent Storm is really, really good.
I would recommend going straight for the standalone expansion though because the base game has some weird ass things going on later in the game. Unless you want to fight Nazis in robots o_o

I would hardly call Hearts of Iron , Europa Universalis, Victoria or other similar Paradox games 'turn based' as they are pausable real time. There are quite few TBS startegy games in Matrix Games' catalog but they tend to be expensive even several years after release so I've not tried any. I don't even know what kind of DRM their store uses if any.
As for free/open source, the ones you could try that come to mind are 1848, Advanced Strategic Command, the free alternatives of civ like games (FreeCiv, FreeCol, Civ-evolution and FreeOrion), Leylines, Steel Panthers : World at War, Steel Panthers : World War II & Main Battle Tank, The War Engine, Titans of Steel: Warring Suns, TribleA and UFO: Alien Invasion. You can find links to them from my site.
From GOG's catalog you could consider following:
Call to Power 2
Disciples 1 & 2
Fantasy Wars
Heroes of Might and Magic® 3: Complete Edition
Jagged Alliance 2
Lords of Magic: Special Edition
Master of Magic
Strategic Command: European Theater
There are couple games at GamersGate like World War 2 Time of Wrath and Elven Legacy Collection. There are others that either have DRM or don't list it unfortunately. You could ask GG staff about them.

Never played it, but there are many positive recommendations about M.A.X. 1.

I can't believe that nobody recommended the wonderful Age of Wonders series yet. :)

+1
Jagged Alliance 2 is a great choice.
Also +1

Master of Magic * 1 million

"They can be commercial, old or new, but should not require online activation and should be available somehow."
sshhh, too early ;-)