Technically speaking ALL openings have an objective other than developing your pieces. For example
Italian Game/Italian Opening goes like so:
1.e4 e5 2.Nf3 Nc6 3.Bc4
From this position you would think it's simply development of pieces however that's not the case. In the Italian game, the opening itself for white is pin pointed on the f7 weakness in black's structure, when black played e5 in response to white's e4 it opened up a weakness in the structure which is the f7 pawn.
That f7 pawn ends up being a pinned pawn in many cases if the player castles on the king side normally, or even sometimes turns in to a bishop sacrifice if you have a good rook lift and an attack going on (however that's too complicated for a beginner in chess). In short Italian Opening's objective is simple, attack the king side's weaknesses and is an aggressive opening most of the time when it goes into the mid game phase.
You also have the Ruy Lopez one of the most famous openings in the history of chess.
The opening focuses on ruining the pawn structure for Black and gaining a potential advantage with the pin on Knight on c6.
You also have the French Defense which is also a famous opening
1.e4 e6 2.d4 d5
The entire purpose and reputation of the French Defense is based on defense and counter play. Usually players of the french prefer a closed game, however that's a conversation for another thread. In the french black's objective is usually counter attack on the queen side while white usually focuses his attack on the king side. French defense also a solid pawn structure which upholds its reputation for defense.
I truly can't give you a list since almost every single opening out there has an objective or a reason for every move. Posting more openings and their objectives + explanation would definitely take too long. However I will leave you with these 3 openings that I was able to mention in this long post and hopefully I helped. If you have any further questions feel free to ask and I will attempt to answer as best I can
Edited part to answer the questions in the comments section :)
King's Gambit (Polerio/Villemson Gambit line is shown below, not the classic line)
1.e4 e5 2.f4 exf4 3. d4
This opening focuses more on offense, with white's move of 2.f4 he is hoping to accomplish mostly 2 things, 1.) ruin the pawn structure (doubled pawns on the f file) and 2. Clear the way to push down the middle. However King's Gambit has 2 variations, Accepted and Denied gambit (accepted is when black takes the f pawn and accepts the sacrifice, Denied is when black continues and ignores the pawn).
If black takes the pawn (which is mostly what white wants) white will in most cases sieze the opportunity and push down the middle with d4 hoping to occupy space and make space for his pieces to enter the game. However in response black usually focuses on the f2 spot weakness where the pawn was sitting before it was sacrificed. Black's objective after the sacrifice is usually have his black bishop hitting that diagonal and unless white can somehow defend it will become a problem (that's why it's called a gambit). From white's point of view however that open f column is also a weapon, it is basically a semi opened column (unless black's f pawn was taken back which would make it a completely open column) for the king side rook. And since King's gambit is usually an open game with bishops this is a dangerous combination.
Here is also the Queen'g Gambit, however instead of me typing the explanation here, you might have a more enjoyable time reading and listening to this
Hope I helped :) If you have even more questions feel free to ask :)