ORGANIC OR INORGANIC? A very brief survey . . .
Organic fertilisers are very safe to use, improve soil structure, and release their goodness over a longer period. Not all organic fertilisers reach Soil Association standards - if you want to be that particular you'll need to do specific research online.
Almost all the usual organic fertilisers derive from animals, though there are a few options for the strict vegetarian. As well as the obvious manure (horse or poultry), there are slaughterhouse by-products such as dried blood, fish blood and bone, and hoof and horn (you can see why a vegetarian wouldn't want to use those!).
Vegetarians can use various seaweed-based products, home-made liquid comfrey extract, or, if they know someone with a micro-brewery, spent hops.
PRO - safe, good for soil structure
CON - hard to be exact, they pong!
Inorganic fertilisers are pure chemicals. They enable the expert gardener to tailor the feeding regime more exactly to the plant's needs at different times in its growth. Though they are often sneered at today as harmful and unnatural, careful use of inorganic concentrates can give excellent results. Gardeners who grow for showing often use inorganics. Remember, most commercial liquid feeds are inorganic.
If you have heavy clay soil you need to be especially careful with inorganic fertilisers, especially "Growmore", as if dry weather follows wet (and when doesn't it?) you can get a build up of strong chemicals in the soil which may harm roots. (The most popular, easy to use alternative is pelleted chicken manure).
PRO - can be used very precisely
CON - soil structure can suffer with long use