I should warn you that the answer will not be short - and I am also prone to giving long replies
Fertilisers and their uses is a rather big subject
Chicken manure is a good fertiliser for virtually most crops - it gives the plants a nitrogen boost which encourages green leaves and growth so good for leafy crops. I tend to put the pellets in the planting hole just before popping the plants in - for hungry summer crops such as squash or sweetcorn, the handful tends to be a bit heavy. However, it is not a general purpose fertiliser but is very useful if the soil is low in nitrogen. The biggest downside with chicken manure is Mr Fox - if there are any in your area, he will dig up your plants in search of the chickens that he thinks you buried .... I've lost a fair number of plants because of using chicken manure.
Now chemical fertilisers are different - you can easily put too much down and it can have the opposite effect. A general purpose fertiliser called Growmore has equal parts of nitrogen (N), potassium (K) and phosphorous (P) and is perfectly fine to use (tends to be the cheapest) Usually a handful over an area, raked in about a couple of weeks before sowing/planting, and then for crops in the ground over a long period or after a particularly wet season, you might need to sprinkle some more round later in the season.
Ammonium sulfate (or nitrate or ureas) is the "pure" nitrogen - it gives plants a huge nitrogen boost. However, too much and you can get "nitrogen burn". Good for leafy vegetables like cabbage, lettuce.
Sulfate of potassium is the potassium element - usually something you put round fruit bushes in spring and add to soil where you grow fruiting vegetables such as tomatoes, cucumber, courgettes and winter squashes. Also good for potatoes and onions. Potash is very soluble so readily leached out of the soil by rain.
Phosphorous is good for healthy root systems - great for carrots and parsnips, but a little for all vegetables will do no harm.
Of the 3 above, a lot really depends on your soil and preparation. If very sandy, nutrients are easily washed out so replenishing them would be a good idea. If you dig in some compost into the soil, you will also help retain water during the (dry) summer months - no nutrient is going to work if the soil is parched. If your soil is clay, then you should ease up on how much fertiliser you put down. Of course, to complicate matters there's also soil pH ... if too acidic, then some minerals such as calcium are not taken up by the plants.
Lime is not a fertiliser, but it does raise soil pH and is especially useful for brassicas (cabbages, brussel sprouts, swede, turnips, cauliflower, broccoli) - I always pop some in a planting hole when planting my cabbages and other brassicas, as my site does have a disease called clubroot which destroys the roots. The raised pH keeps this disease in check.
Don't put lime down where you intend to grow potatoes - the plants will probably be fine, but some varieties are prone to scab which makes the potato skin look corky. The potato itself is edible but sometimes don't store too well if the skin is damaged.
Now - specialist fertilisers are the easy (expensive) route of ready mixed fertilisers and avoids over-fertilising. If there is a high nitrogen level in the soil, plants like potatoes and carrots will put all their growth into foliage and not a lot into the root and tubers. Worse, the leaves grow faster than they should and can make them prone to pests that will take advantage of the tender foliage. So for potatoes, a higher potash and phosphorous with some nitrogen is better. Alliums (onions, garlic, shallots, leeks) are the same - I like to add potash to the soil which helps the bulbs to form, and some phosphorous for good roots. If this sounds complicated, you can buy the specialist fertilisers which take all the guess work out of the equation. Growmore is perfectly adequate for most scenarios.
Now - the "organic" fertilisers are something else. Chemical fertilisers are manufactured or mined, and depending on your sentiments on what this can do to the environment is up to you. Organic fertilisers are obtained from animal sources such as bonemeal, or fish, blood and bone. Organic fertilisers are "slow release", so when you dig it in, there is no fertiliser immediately available to plants and takes a while to break down. Bonemeal will be high in phosphorous and some calcium. Fish, blood and bone is like the organic version of Growmore.
To complicate matters, there are "complete" fertilisers - these contain not only the 3 main elements (N:P:K) but a lot of other minerals essential for growth. Sandy soils may be short on some of these, but clay soils generally do not. You may need magnesium sulfate (Epsom salts) as this can be easily leached out of the soil or if the pH is too low - magnesium is essential for photosynthesis and symptoms of deficiency are seen in leaves which are mottled, pale (there may be other reasons such as lack of manganese but try magnesium first) but watering in a solution can usually solve the problem. Also (going off topic here) - I learned last year that fruit setting on crops such as tomatoes and chilli/peppers are improved if the leaves get an occasional foliar feed of magnesium sulphate solution especially when the flowers appear.
And then there are the home made fertilisers. Compost is a growing medium and some of the plant material that has decomposed will have levels of nutrients available to grow your crops. However, read the label and it will tell you that the plants will need repotting or feeding after about 6 weeks. I always dig compost into soil about a week or so before planting (well, in theory I do but more like sow/plant on the day LOL) For potatoes, I'll dig a trench and fill some of it with compost - my soil is heavy clay so the compost helps to break the soil up, and the tubers when lifted are "cleaner". Potatoes also need a lot of water after the flowering stage so that the tubers will swell - usually at the height of summer when we (sometimes) seem to get a dry spell when it is also hot, so the compost acts a bit like a reservoir too. Now, you can also line the trench with other things too - comfrey leaves are rich in potassium and other minerals, so can be put into the trench at the same time (make sure you don't get the stem or you might end up with comfrey plants taking root) Comfrey tea and rhubarb leaf tea are leaves left to soak in a bucket of water for several weeks/months - be warned that the stench can be horrendous so don't splash it on you. Dilute down to a weak tea colour and use it to feed the plants - maybe once a week (especially if we have a wet summer) or every other week.
I could go on LOL, but I think I've covered the basics. I hope you were able to understand some of it.