Suppose you have a list of MyObject like this:
public class MyObject
{
  public int ObjectID {get;set;}
  public string Prop1 {get;set;}
}
You can do this using the Distinct() method. But since that method uses the default equality comparer, your class needs to implement IEquatable<MyObject> like this:
public class MyObject : IEquatable<MyObject>
{
    public int ObjectID {get;set;}
    public string Prop1 {get;set;}
    public bool Equals(MyObject other)
    {
        if (other == null) return false;
        else return this.ObjectID.Equals(other.ObjectID); 
    }
    public override int GetHashCode()
    {
        return this.ObjectID.GetHashCode();
    }
}
Now you can use the Distinct() method:
List<MyObject> myList = new List<MyObject>();
myList.Add(new MyObject { ObjectID = 1, Prop1 = "Something" });
myList.Add(new MyObject { ObjectID = 2, Prop1 = "Another thing" });
myList.Add(new MyObject { ObjectID = 3, Prop1 = "Yet another thing" });
myList.Add(new MyObject { ObjectID = 1, Prop1 = "Something" });
var duplicatesRemoved = myList.Distinct().ToList();
You could create a custom object comparer by implementing the IEqualityComparer interface:
public class MyObject
{
    public int Number { get; set; }
}
public class MyObjectComparer : IEqualityComparer<MyObject>
{
    public bool Equals(MyObject x, MyObject y)
    {
        return x.Id == y.Id;
    }
    public int GetHashCode(MyObject obj)
    {
        return obj.Id.GetHashCode();
    }
}
Then simply:
myList.Distinct(new MyObjectComparer()) 
You can use GroupBy() and select the first item of each group to achieve what you want - assuming you want to pick one item for each distinct ObjectId property:
var distinctList = myList.GroupBy(x => x.ObjectID)
                         .Select(g => g.First())
                         .ToList();
Alternatively there is also DistinctBy() in the MoreLinq project that would allow for a more concise syntax (but would add a dependency to your project):
var distinctList = myList.DistinctBy( x => x.ObjectID).ToList();