Lay a board across the tops of the two arbor posts. Place a carpenter's level on the board. If the board is level, your posts are standing at equal heights -- which is what you want. Otherwise, make the necessary adjustment.

To ensure that the posts will be plumb (i.e., the vertical equivalent to "level"), nail scrap lumber to each of them to form temporary, stabilizing braces (for a picture of such braces, see photo above). With these braces in place, check for plumbness on all sides with a carpenter's level. If the posts aren't exactly plumb, adjust the positioning of the temporary braces, accordingly.

Garden Arbor Tip

Be careful, when buying your 4x4 posts, to get the height you'll need. Remember, an 8' post sunk 3' into the ground leaves you with an arbor only 5' tall. If you get a deal that you can't pass up on some 8' 4x4 pressure-treated posts, though, it is possible to compensate for their lack of height. Buy galvanized post anchors for each post. Sink the post anchors into the concrete, instead of the posts themselves. The pressure-treated posts would then be attached to these anchors, meaning all 8' would be above ground. Post anchors, however, can be expensive -- make sure that you're not defeating the purpose of getting the lumber at a cheap price!

Securing the Posts With Concrete

Dump a bag of pre-mixed concrete into a wheelbarrow. Add water from a garden hose. Stir the water into the concrete with a shovel, until achieving the right consistency.
Pour this concrete into one of the holes, around the base of the post. Repeat until hole is filled; and repeat for other hole.
Allow this concrete to dry, or "cure" for at least one day.

Garden Arbor Tip

While waiting, lay out the other boards on the ground, make your measurements, and pre-cut and pre-drill as much as possible, to save time.

Since these instructions are meant for non-carpenters, the emphasis will be on the easiest way to accomplish each task....