You'll need to order your garden window several weeks in advance. Garden windows come in a variety of sizes and shapes, but all provide similar elements: extra shelf space and at least four sides of glass, including the top sloped glass.
Order the new window based on the rough opening size of the window you are replacing, not upon the unit size. That means you'll have to remove the old window and be prepared to cover the opening with a temporary cover (e.g. plywood) until the garden window arrives.
After removing the old window, any trim pieces, nails or old flashing, measure the rough opening very carefully. It's also a good idea to check to see if the rough opening is level, plumb and square.
When the garden window arrives, carefully inspect the new window before you install it. Make sure there are no gaps or cracks in the nailing fin. If you have windows that open, check the weep holes at the bottom of these windows. These holes allow water to escape if it happens to get inside the window. To make sure the window is weather-resistant, give it a thorough cleaning with vinegar and water. Check for any warping as you clean.
Even though you ordered the window based on exact measurements, it's important to check the fit of the window; slight variances and gaps are nearly impossible to avoid. Have at least two people dry-fit the window; while one person holds the window in place, the other can go around the window and check for any problems. Use a level to make sure the window is both plumb and level.
Check the edge of the rough window opening to make sure there is no water or structural damage to the sheathing. If there is damage, such as plywood warpage or nail holes, you'll need to repair those problems before you begin the window installation. The ideal is to have a completely clean, unblemished "seal" around the entire garden window.
Place metal flashing (which is just like tape) to the top interior corners of the window. Place the flashing far enough to the outside that it can't be seen from inside the room. Flashing can be found at most home improvement stores and comes in various widths; make sure you get the correct size for your window.
Apply spray adhesive to the flashing and flash around all edges of the rough opening. The adhesive will help the flashing stick and will give you a better barrier. Don't worry about putting too much flashing down. Overlap pieces from the top down. For extra protection on the corners, you can go across them diagonally with the flashing, as in the image.
As you get ready to fit the window into place, apply spray adhesive on top of the flashing.
With a second person helping, fit the garden window into place. Check for level and plumb and tack it into place with nails. Make sure all flashing is covered by the window. Check inside and out for any breaks in the seal.
Once the window has been checked for level and plumb, nail through each hole in the nailing fins. Be careful that you don't hammer the nails so far that they damage the fin, or you may be inviting leaks. For this reason, don't use a power nailer -- sometimes they are too powerful and will tear through the nailing fin.
Nail building paper on any exposed sheathing around the perimeter of the window, and replace the siding you removed. If the siding is vinyl, remember not to drive your nails all the way down because the siding needs to be able to move a little bit.
Now you're ready to trim out the garden window and add finishing touches on the inside.