So, what you're talking about there, Mad, are what are called "contested checks" - this is where two entities are in direct opposition to each other in some way. In those cases, yes, both entities make a skill check with whatever skill is appropriate (they don't need to be the same skill, and usually aren't). The winner, being the one with the higher final number, achieves what they wee attempting, over the other. Important to note: in the case of a direct tie, the status quo is maintained; this means the circumstance doesn't change. So, if you were making an opposed check with someone to hold a door close against them trying to pull it open, a tie would result in the door staying as it is - closed in this case.
Shoving and hiding are the two most common opposed checks outside of conversation.
When you try to shove someone, you are using athletics, but the person you are trying to shove has their *choice* of whether to use their athletics OR their acrobatics to either resit you directly, or avoid the force of your shove.
When you hide, you generally roll a stealth check. In most cases, this should be compared to the *passive* perception of anything that realistically has a chance of perceiving you nearby. If something is actively searching for you, such as by using the search action, then they can make an active perception check against your stealth.
A passive check is your fixed average value for an ability check - it's 10 plus your skill bonus. So, if you are proficient in perception at level 1, and have a +3 from 16 wisdom, your perception bonus is +5; your passive perception, then, is 15.
I should note, that Larian does not seem to understand what passive checks are, and refers to some things as passive checks when it is actually making a roll... if it's making a roll, it's not a passive check.
Other common contested checks happen in conversation - any time your DM invites you to roll insight on a character, you are making a contested check; it will either be against the NPC's persuasion, or deception, in most cases; you won't know which the NPC used (whether they are trying to convince you of genuine earnestness, or trying to deceive you), and the DM generally won't tell you, because that's the point.
To add to Evandir's examples above, a Charisma save is a save that tests your force of presence and sense of self; it's common for resisting attempts to displace or banish, and also for things overtly trying to dominate or force their will or their persona over your own ability to act or speak..
Unfortunately, it shares a certain amount of territory with Wisdom saves, and many things in 5e are Wisdom saves that really *should* be Charisma or Intelligence saves instead... this is a legacy from older editions, where we only had three types of saves (Will, Reflex and Fortitude); 'Will' saves, in those systems covered *everything* related to mental or personal, or just plain non-physical saves... Now that we have three types of mental saves, Intelligence and Charisma should be taking more of that weight, but they didn't get enough to the load shared across to them from Wisdom.
Subtle mental effects - charms and such, should be Wisdom saves.
Overt or forceful mental effects, such as domination, possession and control should be Charisma saves.
Effects that confuse, scatter or disorient the mind or scramble your mental faculties, or that affect memory, should be Intelligence saves.
Out side of that rough rule of thumb, how an effect takes place should also be a consideration; a fear effect might fit as either a wisdom or a charisma save, based on how it is delivered - if the goal is to alter a creature's perception to terrify them without them realising they're being messed with, that should be a Wisdom save. If the effect violently assaults their psyche in an effort to terrify and cow them, that should be a Charisma save.
