When you end your concentration on the spell, the spell ends.
This is explained in the rules on Concentration:
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
If the spell ends, its effects end as well. Everything in the spell description is a spell effect.
This is explained in the rules on Casting a Spell:
Each spell description begins with a block of information, including the spell's name, level, school of magic, casting time, range, components, and duration. The rest of a spell entry describes the spell's effect.
In the case of fog cloud, one of the effects is that there is "a 20-foot-radius sphere of fog centered on a point within range." With the end of your concentration, there is no longer a 20' radius sphere of fog. Another effect of the spell is that "..its area is heavily obscured." With the end of the spell, the area is no longer heavily obscured. Both of these effects are ended immediately when the spell ends.
What happens next is up to the DM
This is explained by Rule 1, "The DM describes the environment"
The DM tells the players where their adventurers are and what’s around them, presenting the basic scope of options that present themselves (how many doors lead out of a room, what’s on a table, who’s in the tavern, and so on)...[and moving into Rule 3]...Often the action of an adventure takes place in the imagination of the players and DM, relying on the DM’s verbal descriptions to set the scene.
As a DM, if you want some ambience from a spell to linger, great! Strong descriptions of lingering effects add verisimilitude and atmosphere. As a DM, if you want all traces of a spell to immediately vanish, great! Asserting that it is 'magic' and not 'real', will emphasize how powerful casters are, in imposing their will on a recalcitrant universe.
If you do choose to have lingering effects, though, I would strongly suggest that you either make these non-mechanical, or be scrupulously consistent in applying them. For example, the turn after a fog cloud ends, you will not have "a 20-foot-radius sphere of fog centered on a point within range." But you might want to describe how there is now a 'wet and sparkling mist in the general area, and a light dew on the objects where the cloud was'. That is purely atmospheric and shouldn't cause problems.
The turn after a fog cloud ends, you will not have "its area is heavily obscured." But it might be tempting to have 'half its area lightly obscured' as a lingering vestige of the spell effect slowly resumes normalcy. The danger in a lingering condition having a mechanical effect is that your players may come to rely on that, and be frustrated if you do not always apply it. How come the evil druid gets an extra turn on his fog cloud and we don't? But you didn't do that last time! How fast does the wind need to go to disperse the cloud immediately? If you set up the lingering effects so that they matter, expect the players to be invested in the outcome and to react accordingly.