Nanotechnology is technology that operates on nanometre (a billionth of a metre) scales. Since a nanometre is just one order of magnitude above (ten times larger than) an Angstrom (a measure of distance used on the scales of atoms), a civilization that develops nanotechnology is able to start manipulating matter at the atomic level.
The most important impact that the discovery of nanotechnology by a civilization is the ability to build nanorobots (sometimes referred to as 'nanobots' or 'nanites'). These have a multitude of uses as well as vulnerabilities.
One of the first uses that civilizations usually find for true nanorobotics is in medicine, specifically in localised drug delivery. This basically consists of specially designed delivery spheres that are transported around the body in the blood, chemically bonding to and entering certain types of cells in certain areas of the body, where they break up and release their drug. If a poison is used, the system can be used to target and kill cancerous cells or, in a more sinister use, kill an organism using microscopic quantities of poison targeted to a specific part of the body (neurons, cells of a vital organ).
These medical nanorobots are variably referred to as protocells, nanospheres, drug delivery systems, artificial viruses etcetera, and are not always recognised as nanorobots in their own right. They are relatively simple and easy to design, requiring no power supply, propulsion system or moving parts; as such these (or something similar) are usually the first form of nanorobot to be developed by a civilization, if they are classified as nanorobots at all.
The next step up from medical nanorobots is programmable matter, though this particular technology usually stems more directly from integrated circuits than medical nanorobots. Programmable matter is made up of individual units sometimes reffered to as claytronic atoms (though it should be made clear that these units are not atoms in themselves), which are miniscule, usually spherical (as opposed to the less advanced wafer design), integrated circuits which have the ability to send and receive signals, as well as specifically control their electrical charges.