The Remote Sensing Tutorial written by N.M. Short (and viewable online, see
below) provides the following fairly formal and comprehensive definition of
applied remote sensing:
The acquisition and measurement of data/information on some property(ies) of a phenomenon,
object, or material by a recording device not in physical, intimate contact with the feature(s) under
surveillance; techniques involve amassing knowledge pertinent to environments by measuring force
fields, electromagnetic radiation, or acoustic energy employing cameras, lasers, radio frequency
receivers, radar systems, sonar, thermal devices, seismographs, magnetometers, gravimeters,
scintillometers, and other instruments.
The same tutorial also provides this shorter (more colloquial) version: Remote Sensing involves the detection and measurement of photons of differing energies emanating from distant materials, by which these may be identified and categorized by class/type, substance, and spatial distribution.
A Geographic Information System (GIS), as defined by the Association for Geographic Information, is A system for capturing, storing, checking, integrating, manipulating, analyzing, and displaying data which are spatially referenced to the Earth.
For additional definitions, see the "ASPRS Fact Sheet" put out by the ASPRS: The Imaging & Geospatial Information Society, of which I am a member.
One needs more than just definitions though. To get more of an idea, check out these various webpages.