Differences between object initializer and constructor?

Object Initializers were something added to C# 3, in order to simplify construction of objects when you're using an object.
Constructors run, given 0 or more parameters, and are used to create and initialize an object before the calling method gets the handle to the created object. For example:
MyObject myObjectInstance = new MyObject(param1, param2);

In this case, the constructor of MyObject will be run with the values param1 and param2. These are both used to create the new MyObject in memory. The created object (which is setup using those parameters) gets returned, and set to myObjectInstance.
In general, it's considered good practice to have a constructor require the parameters needed in order to completely setup an object, so that it's impossible to create an object in an invalid state.
However, there are often "extra" properties that could be set, but are not required. This could be handled through overloaded constructors, but leads to having lots of constructors that aren't necessarily useful in the majority of circumstances.
This leads to object initializers - An Object Initializer lets you set properties or fields on your object after it's been constructed, but before you can use it by anything else. For example:
MyObject myObjectInstance = new MyObject(param1, param2) { MyProperty = someUsefulValue

};

This will behave exactly the same as if you do this:
MyObject myObjectInstance = new MyObject(param1, param2); myObjectInstance.MyProperty = someUsefulValue;

However, object initializers are simpler to read (especially when you set multiple values), so they give you the same benefit as many overloads on the constructor, without the need to have many overloads complicating the API for that class.