Earthbound was a video game for the Super Nintendo Entertainment System.
Mother 2 was a video game for the Super Famicom.
Really, Mother 2 and Earthbound are the same game.
They take place in "Eagleland", a parody of the kind of 1950's Leave It To Beaver-esque America (except The Beatles are already big). Oh, it is quite hilarious.
This is Ness, the male lead and main character. You control Ness, but you aren't Ness (your you, but this never comes up until the very end of the game)
Something bad is happening. It isn't 100% clear what it is, but it's definitely bad. A meteorite strikes near Ness's house. You go to check it out, but the path is blockaded.
Back home Ness's neighbor Pokey, above, gets Ness to help him (Pokey) look for his (Pokey's) brother Picky. Ness, his dog King, and his neighbor Pokey sneak out to check out the meteorite. Pokey is cowardly and mean (this becomes important later), but King and Ness fight off snakes and crows and eventually gets up to the meteorite where picky is. An alien fly with psychic powers named Buzz Buzz comes down from the heavens to try to stop the bad things that are happening. Soon after this Ness, Pokey, Picky and Buzz Buzz are attacked by a different alien Starman Jr.
Buzz Buzz defeats Starman Jr. and the whole group goes to Pokey's house, where the P-pair (Pokey and Picky) get in trouble for sneaking out. Their mother slaps the fly in the air, killing Buzz Buzz. Buzz Buzz explains that you have to visit 8 sanctuaries to prevent a cosmic horror from eating the earth.
After that it gets a little odd, in funny and occasionally meta-fictional ways. A good explanation of a few of the subtler touches is available here. He gives away the ending of the game though, so don't read if you intend to play. Also, don't attempt to play without some sort of guide. The game was designed to be play with a guide.
A few stand out devices:
Ness can buy a house, which serves no purpose except to be a house
There's an optional quest that has no reward except a warm feeling inside
A traffic jam blocks a road, and then clears up (normally in video games things only happen by player action, making this almost disturbing)
Two bosses are unbeatable
Lots of music, images, and plots are only scene once and vanish (MacGuffins)
The Blues Brothers
Items that can only be used one obvious time (with a bizarre second twist if it isn't a McGuffin)
Anyway, this cosmic horror (called Giygas) has already infiltrated the world to a great extent, causing people, inanimate objects, and animals to go crazy. In addition to aliens and wild animals, on the way to defeat evil Ness will be attacked by:

Zombies

Abstract Art

Annoying Old Party Men

A cult leader obsessed with the color blue (named Carpainter!)

Loaded Dice
And many, many others. Fortunately Ness won't have to do it alone, Ness will be aided by at most three people at a time:

Poo, Jeff, Ness, and Paula
Also, the Blues Brothers (well, copyright friendly analouges) help you at several points.
Eventually you get to the end and discover that Giygas sacrificed his living form to be beyond time and invincible. Using the technology designed by:

Jeff's father, Dr. Andonuts

The Apple Kid

And the Mr. Saturns (a possibly alien race)
The four of you have to give up your bodies, enter robots and travel to the end of time to fight Giygas. Pokey, who has been manipulating people for Giygas for some time now, appears in a mecha-spider suit to kill you. Giygas had sacrificed its own body for omnipotence, and dwells inside the familiar looking Devil's Machine to keep from destroying reality.

Pokey has taken control of the machinations of Giygas (taking control of the Starmen, several large corporations and the entire criminal world) and doesn't intend to give them up. The four of you, at this point armed with powerful weapons and great psychic powers attempt to wrangle Pokey.
Pokey opens the devil machine, and the blind idiot god Giygas emerges.

Giygas emerges from the devil machine a swirling red mass that undulates between being a skull and a fetus:

Then things happen that someone who has not played the game can understand. Lets just say that beating the blind idiot god is impossible, but you still have prayer!

Since this post is an obscure game on a dead console, it is not likely that you will ever play it if you have not. Suggesting that you play a
ROM on an
Emulater just wouldn't be right. So Loth Beg would like to recommend that you watch The Blues Brothers, the film version of magical realism, and celebrate their important role in Earthbound.