To shlep is to drag or lug something. If you have to pick up your dog and carry him up the stairs to your apartment, you can say that you need to shlep him.
When you shlep a heavy box down the street, you awkwardly pull it, or you carry it with difficulty. You can also describe your trip along the sidewalk with the heavy box itself as a shlep: "It was such a shlep to get that box over to Adam's house." You can also spell the word schlep, and it comes from a Yiddish root, shlepen, "to drag."