Of all neem seed extracts the neem seed oil is the best known and most popular.
Unfortunately making pure neem seed oil requires considerable knowledge, skills and equipment.
Luckily there are other ways to extract the active substances from neem seed kernels. Organic solvents work best, that's what is used most in industrial neem extract production. At home you can use either an easily available organic solvent (ethanol or methanol), or, even simpler, use water!
Yes, watery neem seed extract is quite effective. Most of the active ingredients in neem seed kernels are not very water soluble, but because they are so concentrated in the seeds even a watery extract will be very powerful.
A word of warning: neem seed oil and neem seed extracts have a very strong, pungent smell!
There is a way to make your own neem seed oil. It will not give you high yields, and it will not be of the best quality. But it will give you some oil.
To make a strong enough extract with water you need about 500 g of seed kernels to 10 l of water. (Of course you can adjust the overall amount as required.)
Alternatively you could just put the bag inside a bucket or tub and steep it over night, but the above method will give a more powerful extract.
The extract can be used as is. However if you want to use it for spraying you will need to filter it first or it will clog up your sprayer.
This by the way is the most common method of using neem in agriculture in third word countries. 20 - 30 kg of kernels usually treat about one hectare. That means you need about 50 liters to treat 1000 m2.
Azadirachtin and the other limonoids (the substances responsible for neem's insecticidal properties) are highly soluble in alcohol. Alcoholic extracts are over 50 times as concentrated as watery extracts.
To make an alcoholic extract you soak the kernels in either ethanol of methanol and then strain and filter the brew. The concentration of the resulting extract can vary greatly, from 0.2% to over 6% active ingredients.
Unless you purify the crude extract, and adjust the concentration and the pH, the solution will not be stable. The active ingredients in neem usually decompose rapidly.
I would say leave the oil pressing and alcoholic extraction to the big processors and buy pure neem seed oil and extracts, of known concentration, from reputable, organic neem product suppliers.
For home use and home remedies stick to the watery extracts, which you can easily prepare freshly as needed.