Both basses and electric guitars are variations of the same instrument--the guitar. The most basic difference between the two is that the bass guitar plays notes in a lower register than the standard electric guitar. While many variations of both the bass guitar and the electric guitar exist, the basic differences are the same no matter what types of basses and guitars you are examining.
An electric guitar usually has six strings while a bass only has four. Many variations of these instruments exist, but the six-string guitar and the four-string bass are far and away the most common types of these instruments. Because the bass plays deeper pitches than the guitar, bass strings are significantly thicker than guitar strings. The thickness of bass strings also makes them much more expensive than regular guitar strings.
The bottom four strings of a guitar are tuned to the same notes as the four strings of a bass (E, A, D, G), but a bass guitar is tuned exactly one octave lower than a guitar. Because of this, guitar music is written on the treble (G) clef while bass music is written on the bass (F) clef.
Bass guitars tend to be slightly bigger than regular electric guitars. This is because the thicker strings exert greater force on the body, neck and hardware, resulting in a bigger body (usually), a thicker neck and much larger hardware (bridges and tuning pegs). Both bass guitars and standard electric guitars come in a wide variety of shapes, sizes and colors.
The bass guitar (along with the drums) is generally considered part of the rhythm section of a band while the guitar is often considered a part of the lead section. This isn't to say that the bass doesn't take leads from time to time or that the guitar isn't often used to play rhythm, but rather that these are the traditional roles of these instruments. Bassists tend to play single note lines while guitarists play chords more often than single note riffs.
Bass guitar amplifiers are different than guitar amplifiers. Most of the time they can be used interchangeably, but bass amps are made specifically to accentuate the low end of the bass guitar. Guitar and bass effects pedals are similar to amps in this regard. Both guitarists and bassists can play with picks (although many who play both instruments prefer playing with their fingers), but because of the thicker strings, bassists have to use thicker picks to prevent them from breaking.