get and set the checked state of checkbox from another checkbox in another form

When you click the button or something to open the Form1, place following code on that location.

Form1 f1 = new Form1();
f1.checkBox1.Checked = !checkBox2.Checked;
f1.Show();

This is enough to do what you want.

Currently you create a new object everytime you check/uncheck the check box, and I am sure you must be creating a new object where you show the form too.

[Edit]
If form is already open or you don't want the form to open at all; then put your declaration of form Form1 f1 = new Form1(); at the top of class to get the single object of the form.