I have an Ivy Lane Denizen, it says that he may put a +1/+1 counter on target creature whenever a creature enters the battlefield under my control. Does the counter go on Ivy Lane Denizen specifically or on the creature that just entered? Or does "target" mean I choose any creature of my choice?
Cards in Magic: The Gathering (at least when looking at the official Oracle wording, found on Gatherer) usually mean exactly what they say, nothing more and nothing less. If you had to place the counter on the denizen, it would've said "Whenever another green creature enters the battlefield under your control, put a +1/+1 counter on Ivy League Denizen.", and if you had to put the counter on the creature that just came out, it would've said "Whenever another green creature enters the battlefield under your control, put a +1/+1 counter on that creature."
When an ability or spell you play uses the word "target", you are entirely free to choose a target to your liking, within any explicitly mentioned restrictions (it has to be a creature since it says "target creature", said creature can't have shroud or protection from green, etc.)
You can put the counter on any creature you desire. So it can be Ivy Lane Denizen, creature that entered the battlefield, one of your other creatures or even one of your opponent's creatures. You need to be able to target that creature, though.
"Target" means that you get to choose what object is going to be affected by your spell or ability. You make this choice when you cast the spell or when the ability is activated or triggered.
"Target creature" means that you must choose a creature.
From the MTG Basic Rulebook:
Casting a spell
If the spell is an instant or sorcery and it has a target , you choose what (or who) that target is.
When you see the word “target” on a spell or ability, you have to choose one or more things for the spell or ability to affect. You’ll be able to choose only certain kinds of things, such as “target red permanent” or “target creature or player.”
You choose the targets for a spell when you cast it, and you choose targets for an activated ability when you activate it.
The rules for selecting targets for a triggered ability, such as Ivy Lane Denizen's ability, work the same as for casting a spell or activating an ability.
Magic is a game where wording matters a lot. You found two very important and common wordings in cards in this one card we are talking about.
Whenever another green creature enters the battlefield under your control, put a +1/+1 counter on target creature.
Let's dig in as this will help you understand other rulings in the future.
The first word that matters to you here is "When", this gives you an indication of a triggered ability, something happens when something else happens.
Now we know that a trigger happens, what kind of trigger is it? "enters the battlefield". So now we know what will cause it to trigger, when something enters the battlefield. but we need to know what the something is. So the condition for this ETB (enters the battlefield) trigger is "another green creature". The creature has to be green and has to be another, so it means that this won't trigger when Ivy Lane Denizen itself enters the battlefield, at least for its own triggered ability.
Now your question is about the targetting part of the ability.
There is quite a bit you can read about targetting on the comp rules but in this case use the same logic as we used for the ETB side. So we know what causes it to trigger, now we need to know what effect the trigger brings. The trigger makes you target something. The something in this is creature. And what modifier condition is there for creature? None. In this case, in this card, the only condition for the target is that it has to be a creature (and the target be legal). if, just like the ETB it said "another" creature, then this would be the modifier for the condition.
Mostly, rules on MTG cards are as simply as the text in the card, there is no need for interpretation. Always use the gatherer for clarification, but I would suggest trying to find the solution yourself first.