You can use whatever damping function you feel appropriate, but generally, you would repeat:

1. figure out which way to move
2. figure out and damp how much to move to keep the same screen position
3. move

If you threw it into a coroutine with a wait at the start, you could add inital delay and other things, but generally that's it.

Here is a simple damp function that will do generally what you need using Vector3.SmoothDamp:

var dampTime : float = 0.3; //offset from the viewport center to fix damping
private var velocity = Vector3.zero;
var target : Transform;

function Update() {
    if(target) {
        var point : Vector3 = camera.WorldToViewportPoint(target.position);
        var delta : Vector3 = target.position -
                    camera.ViewportToWorldPoint(Vector3(0.5, 0.5, point.z));
        var destination : Vector3 = transform.position + delta;
        transform.position = Vector3.SmoothDamp(transform.position, destination, 
                                                velocity, dampTime);
    }
}

It's been written to work for both orthogonal and perspective cameras. If you want, you can apply some optimization specifically for orthogonal because you wouldn't need to calculate viewport points for an orthogonal camera.

Here is Scott's answer in C#

using UnityEngine;
using System.Collections;

public class SmoothCamera2D : MonoBehaviour {

    public float dampTime = 0.15f;
    private Vector3 velocity = Vector3.zero;
    public Transform target;

    // Update is called once per frame
    void Update () 
    {
       if (target)
       {
         Vector3 point = camera.WorldToViewportPoint(target.position);
         Vector3 delta = target.position - camera.ViewportToWorldPoint(new Vector3(0.5f, 0.5f, point.z)); //(new Vector3(0.5, 0.5, point.z));
         Vector3 destination = transform.position + delta;
         transform.position = Vector3.SmoothDamp(transform.position, destination, ref velocity, dampTime);
       }

    }
}

This is probably not needed at this point but just in case someone stumbles upon this answer, this is the modified code needed to lock the Y-Axis of the camera, just like an old school platformer. All original code by Scott Kovacs.

//Written by Scott Kovacs via UnityAnswers.com; Oct 5th 2010
//2DCameraFollow - Platformer Script

var dampTime : float = 0.3; //offset from the viewport center to fix damping
private var velocity = Vector3.zero;
var target : Transform;

function Update() {
    if(target) {
        var point : Vector3 = camera.WorldToViewportPoint(target.position);
        var delta : Vector3 = target.position - camera.ViewportToWorldPoint(Vector3(0.5, 0.5, point.z));
        var destination : Vector3 = transform.position + delta;

       // Set this to the Y position you want the camera locked to
        destination.y = 0; 

        transform.position = Vector3.SmoothDamp(transform.position, destination, velocity, dampTime);
    }
}

I would like to add, if rigidbody object you folow moves not smoothly (jittery), even if camera moves smooth. Turn on Interpolation in properties of rigid body. I couldn't figure out this for months (

You can read more about this here: http://docs.unity3d.com/Documentation/ScriptReference/Rigidbody-interpolation.html