Hydroponic means growing plants in a nutrient solution. Hydroponics started over 100 years ago in man's quest to beat the elements and secure food supply. However William Frederick Gericke is credited with the term "hydroponics" in 1937 when he grew 7.6m high tomatoes in nutrient solution. 

Today Netherlands, USA and Israel provide the thought leadership and innovation in the world of farming.  Japan, China, Singapore are also innovators in their own right.

When we setup our farms, we made visits to Israel to study their methods, got in a consultant from Netherlands, and much of our equipment is still of US-Netherlands origin e.g. the Hoogendoorn fertigation and climate control system.

Much has changed since. Today India not only makes such equipment, but more importantly India has a ready pool of experts who are building low-cost alternatives and selling them commercially.  

As a commercial grower we select the crop first, and then the grow system.  

For example, lettuce is best grown in deep water culture (DWC), then Nutrient Film Technique (NFT), and only if there is no choice or there is excess capacity would we use coco-peat troughs or dutch buckets.  Spinach is the reverse, its best to grow in coco-peat.

As a general rule (with many exceptions) look at the root system and identify if the crop is single-cut or multi-cut.  You don't want to expose a multi-cut plant with fibrous roots to a DWC tank. The reason is the roots will eventually start to 'melt' or succumb to root rot. Hence a single-cut is best suited to DWC, not a multi-cut.

With vines, its best to grow plants that are susceptible to fast travelling diseases like bacterial wilt in separate troughs, not running troughs.  You may even consider dutch buckets if your budget allows.

There are of course many other methods like ebb and flow, but we are covering just the major ones.

As a hobbyist or for home use, Kratky for vines, and NFT for leafy vegetables are the most convenient systems to setup. They are also space saving and easy to maintain. 

Just as we humans require vitamins, minerals, amino-acids etc, so do all plants.  Spinach for example gives us our required daily amount (RDA) for potassium, iron, folate etc.  If we don't get it in our food, we buy supplements or multi-vitamin and other combos.

Plants have similar needs for minerals and are typically given this through fertilizer either as a nutrient solution or as foliar spray. 

As a hobbyist or for own use, simply buy a standard pack of 2 or 3 solutions and apply.  Typically they are labelled solution A, B or A, B and C.  Reason for them being separate is some minerals react with one another, so need to be mixed in water first, then combined.  Miracle Gro is the world's largest and most well respected hydroponic solution brand, though of course there are many other providers.

As a commercial grower, its practical to buy basic minerals, and combine them in proportions suitable for individual crops and their life-stage.  

A number of bio-stimulants are also used eg epsom salt, humic acid, fulvic acid, seaweed extract, etc - but these are typically for more sophisticated growers and agronomists.

Under-dosing, e.g below EC 0.7 for lettuce,  creates weak plants with sluggish growth and exposes them to disease, while over-dosing results in plant distortion or even death, so solution PH and EC must be monitored and controlled within the narrow band define for each plant such that mineral absorption is maximised. Root inspection is also a must.

Flushing nutrient solution is done by commercial growers based on a schedule. For home use, I suggest every 3 weeks for a start till you become familiar with plant behaviour under different conditions.

I use hydroponic nutrients for soil - based plants for added vigour, but it is generally cheaper to use slow-release nutrient balls from the market over the long run