Passive
If you've ever taken a cutting from a houseplant and placed it in a jar of water to establish roots, you've used a passive hydroponics method!  With no moving parts, passive systems are great for beginners because they are inexpensive, portable and simple to set up and maintain.

Active
Active systems, on the other had, employ pumps and other devices to cycle and deliver the nutrient solution to the root zone.  Here are some of the preferred types of active techniques;

Deep Water Culture ( or Continuous Aeration Technique:  With this self-contained method, plants are suspended above the water level and a submersible pump is used to constantly bathe the roots in nutrient solution.  As plants mature, the roots will grow into the continuously circulating reservoir.

Flood and Drain (or Ebb and Flow):  For this approach, plant roots are intermittently flooded with nutrient solution.  The frequency and duration of the flood depends on factors like size and type of plant and growing media; a typical flood cycle lasts for 15 minutes and occurs every 2-4 hours during the day.  Roots are nourished and aerated as the cycle repeats.

Drip:  This is a substrate systems where a pump delivers solutions from a main reservoir to drip emitters positioned at the base of each plant via individual supply lines.  Depending on the growing medium, some will drip continuously, others are set on a timer to drip 15 minutes every 2-4 hours during the day.  Drip systems that use a stonewool medium give you the most "margin of error" as they retain water incredibly well.

Aeroponics:  In aeroponics, plants are suspended without the use of a growing medium, and their roots are continuously sprayed with a fine nutrient - and oxygen - rich mist.  Because roots have unlimited access to oxygen, water and nutrients, the entire plant can grow at a phenomenal rate.  These systems have a small margin of error and are recommended for more experienced gardeners.  Delicate sprayer nozzles must be kept free of debris as they can clog easily, and equipment or power failure can cause total crop loss very quickly.  Popular for cuttings and fast harvesting plants.

Nutrient Film Techinique (NFT):  A bare-root system in which nutrient solution is constantly pumped over plant roots at a depth of 1/4" to 1/2" to form a thin film of nutrient, giving roots access to nutrient and air simultaneously.  The solution cycles between the main reservoir and the grow channel (or gulley) which is tipped at a slight angle to create the desired film effect and prevent roots from "damming" the channels.