All brewing, from commercial manufacturers producing millions of pints a year down to homebrewers like us, follows this same process.
The only real differences are whereabouts in the process you start. All commercial brewers and many homebrewers use the All Grain (sometimes known as Full Mash) method which means that they start with just malted barley and water and take full control of every part of the brewing process from that point on. They heat (mash) the malt and grains to make the sweet wort, boil this with the hops to make the bitter wort and finally ferment this with the yeast to produce the beer.
Malt extract brewers miss out the first step and buy a sweet wort which has already been produced by a commercial brewer. This wort is provided either with some of the water removed as a thick syrup called Liquid Malt Extract (LME), or with all of the water removed as a dry powder called Dry Malt Extract (DME). Malt extract brewers start by adding water to the malt extract to recreate the sweet wort and then following the rest of the process – boiling it with the hops, then fermenting it with the yeast.
Kit brewers miss out the first two steps and buy commercially-produced bitter wort to which the hops have already been added and boiled. This, again, is provided in the form of thick syrup which is rehydrated with water and some form of extra fermentable material (often 1Kg of standard table sugar) added before adding the yeast and fermenting it as normal. Sometimes referred to rather rudely as “kit and kilo” brewing, it is nonetheless possible to make perfectly acceptable beer this way, although “two-can” kits (which don’t need additional sugar) will usually produce better results.
The method (or methods) which you choose will be based on what you personally want to achieve and how much time and money you want to spend. If you want to have full end-to-end control over your brewing they you’ll need to go down the All Grain route, but this takes longer, is more complicated and requires rather more in the way of equipment. If you’re prepared to let someone else mash your grains for you then you can brew with slightly more expensive malt extract (you’re paying for someone else to have made it). If you just want to make drinkable beer in the quickest and easiest manner, with a minimum of equipment and fuss then kits may be the way to go (but also the most expensive).