I am sort of confused as to which city is who's stronghold. Apparently the map from Bethesda showed that Solitude belonged to the Imperials while Whiterun belonged to the Stormcloaks. Now I have read that Whiterun is the capital of Skyrim and that Stormcloaks are being executed there.

Can someone clarify which is the stronghold of the Legon/Empire, and which is the stronghold of the Stormcloaks/Rebels?

Whiterun is the capital of Skyrim. Solitude is where the Empire's forces reside. Windhelm is where the Stormcloak rebels reside.

Oh, I must have gotten Whiterun and Windhelm mixed up there.

Why doesn't anyone seem to want to capture the capital though?

EDIT: Sorry, forgot to thank you for helping me to clear my bad mix-up! So thank you!

I believe all the Jarls of each hold are supposed to be persuaded by the player to join either side. Kind of like how the Champion of Cyrodiil had to go around destroying Oblivion gates for all the counts and countesses. But don't quote me on that...

according to UESP Solitude is the capital of Skyrim, as it was the seat of the old High King.

All I know is that it was the capital in the time of Oblivion.

I'm pretty sure Solitude is capital.

Solitude was the old capital, Windhelm is the current capital.

The capital changes depending on who is High King. Windhelm was once the capital. Winterhold was once the capital, as was Whiterun.