A great and easy way to respect the environment while hiking is to remain on the trail at all times. If you stay on the trail you don't disturb the beautiful nature that surrounds you.
Trails are there for a reason; use them! Though it is tempting to create your own shortcut or cross a switchback, don't do it. When you don't remain on the trail you cause the rest of the area to erode.
Staying on the trail is typically rewarding; it can be easy to get lost or disoriented as soon as you leave the safety of the trail. It is also usually the best way to get to where you are going. If you leave the trail you might be left crossing that stream on your own, without a bridge. That can be a dangerous and very cold experience depending on where you are! Staying on the trail is also better for the environment, leaving the areas off the trail untouched and undisturbed.
Try "leave no trace" hiking and camping. Whatever you bring in to a hiking, camping, or backpacking trip you bring out. Leave minimal impact every time you hike. Volunteer to help rebuild trails with your local park rangers. Be pro-active and help educate others on the benefits of remaining on the trail. Familiarize yourself with the park rules and oblige by them. If they say dogs must be on-leash or no motor vehicles, don't push your luck by letting Fido run free or revving up the ATV. Those rules are there to protect you and your pets.
Even if you aren't a hiker this same concept applies to all of us. Where there is a sidewalk or path, opt to take that sidewalk or path. Don't cut across your neighbor's grass to get to your yard.
Read about Stay The Trail Colorado to get more ideas on how to help preserve your local trails.
Read up on hiking etiquette before your next trip!