When the grey American squirrel was first introduced to Britain, it occupied the same niche as the British red squirrel. In most situations, the invading squirrel out-competes the native squirrel and gains dominance over the niche.

http://www.nebiodiversity.org.uk/biodiversity/species/mammals/redsquirrel/d...

There are many species of Caribbean anoles, and they all occupy the same area. To avoid competition for a common resource, the anoles occupy different areas of the forest (ground, canopy, tree trunk, etc.) so they can all benefit from the resources available.

http://www.uic.edu/classes/bios/bios101/interactio/tsld016.htm

Leopards and lions both eat the same prey: zebras. Because this is not an unlimited resource, the leopards and lions enter into interspecific competition.

Ducks capture fish to eat, making ducks the predator and fish the prey. This is classified as predation, where one animal benefits and the other is harmed by being eaten.

Monarch butterflies partake in the nectar of the milkweed plant. Since milkweed plants are neither harmed nor helped from this interaction, this is a commensalistic relationship.

The oxbird gets both a ride and a snack on the hippopotamus when it lands on its back to eat the bugs. The hippopotamus gets a free cleaning service when the ox-birds eat the bugs off his back. Both species benefit, so this is a mutualistic relationship.