How do I make my player jump at a fixed height, and only once until he touches the ground again?

Well first off you would want to add a rigid body to your character ( this helps with physics such as gravity and so on)

Then the code would look something like this

 public float moveSpeed;
 public float jumpHeight;
 public float moveVelocity;
 
 bool grounded;
 
 //This is for 2D not 3D, I am just taking a wild guess this is a 2D game
 private RigidBody2D rigid;
 
 void Start()
 {
 
 //Get the rigidbody of character 
 rigid = GetComponent<RigidBody2D>();
 
 //Set grounded to true or false (Depending if your character starts in the 
 //air
 grounded = true;
 }
 
 void Update()
 {
 
 
 moveVelocity = 0;
 
 // If the character moves right 
 if (Input.GetKeyDown(KeyCode.D))
 {
 
 moveVelocity = moveSpeed;
 
 }
 // If the character moves left;
 if (Input.GetKeyDown(KeyCode.A))
 {
 
 moveVelocity = -moveSpeed;
 
 }
 
 //If you press spacebar and your are on the ground then you can jump
 if (Input.GetKeyDown(KeyCode.Space) && grounded == true)
 {
 //I am not to sure about the rigid.velocity code play around with it
 //I think in 2D you can set it to a Vector2 instead of Vector3 otherwise just 
 //change it
 rigid.velocity = new Vector2 (moveVelocity, jumpHeight);
 
 //After your character jumps grounded has to change to false since the 
 //character is not on the ground anymore
 grounded = false;
 
 }
 
 // If your character is not falling nor jumping then change grounded true
 //Since he has landed.
 if (rigid.velocity.y == 0)
 {
 
 grounded = true;
 
 }
 
 }

Since I am not near unity I am not sure of the spelling of all the code I written but it should be more or less correct. I added comments in the code to explain each part.

Also place the your movement script onto your character (Otherwise RigidBody2D won't know what your trying to code)

Hope this help