Home brewers have three methods available to them for making beer. They are brewing kits, malt extract brews and all grain brewing otherwise known as full mash brewing. All three methods are described below.

Brewing Kit 
Method - The simplest method for creating your own beers at home. The kit contains a concentrated version of the finished wort, the mixture of grains and hops, and yeast so only water needs to be added. Once the beer has been fermented you can either bottle the end product or cask your beer to enjoy. This method causes the least amount of mess and requires minimal effort.
Equipment and Ingredients - The kit you will require for this method of home brewing is also minimal and only a fermentation vessel, stirring spoon, hydrometer and thermometer or temperature strip is necessary. Cask or bottles.
Instructions - Follow the instructions provided by the manufacturer on the kit.

Malt Extract
Method - This is slightly more advanced that the brewing kit method. The malt extract approach skips the mash but allows you to add the hops yourself so that you can start to tweak and make your own recipes while still keeping it relatively simple and mess free.
Equipment and Ingredients - Hot liquor tank, fermentation vessel, stirring spoon, hydrometer, thermometer or temperature strip, malt extract, hops, yeast and water. Cask or bottles.
Instructions - Sterilise all of your equipment, pour the volume of water required into the hot liquor tank and heat. If the recipe calls for steeped grains heat the water to 70oC, add the grains to the grain bag and infuse for 30 minutes at 65-70oC. Strain the grains and remove the bag before bringing to the boil. If no steeped grains are required bring the water to the boil. Remove from the heat and add the malt extract and stir to dissolve. Bring back to the boil and add the hops as per the recipe. Cool the wort and transfer to a fermenter. After fermentation is complete bottle or cask as per your preference.

All grain / Full mash
Method - This method requires an in-depth knowledge of brewing has multiple processes and steps as well as requiring some specialist equipment e.g. mash tun. The time taken for this method is 4-6 hours. Below is a basic description of the equipment, ingredients required and the method for making your own beer. More extensive descriptions are available on other websites and in homebrewing books.
Equipment and Ingredients - Hot liquor tank, mash tun, sparging arm, boiling tank, wort chiller, fermentation vessel, hydrometer, stirring spoon, fermentation vessel, cask or bottles. Grains, hops, yeast, sugar, water and finings if you don't want some haze to your beer.
Instructions - Sterilise all of your equipment, pour the volume of water required into the hot liquor tank and heat to 70oC. Pour the required volume for the mash into the mash tun, add the grains and mix to ensure no dry patches. Leave for 60-90 minutes at 66oC before sparging with water at 70oC being careful not to disturb the grain bed. Transfer the wort to the boiling pot, check the density of solution with the hydrometer and boil adding the hops and finings (if you want them) as per the recipe. Chill the wort using a wort chiller or sink filled with water and ice. Once cooled to ~22oC strain out the hops and transfer to the fermentation vessel check the original gravity and adjust with water or sugar as required. Pitch the yeast and leave to ferment. After fermentation prime the wort, this will carbonate your beer by the addition of sugar to restart fermentation in your storage vessel. Store the finished product for 2-4 weeks to allow the beer to carbonate and then enjoy the finished product.

Now you know of the three methods available to you, I hope you feel inspired to go out there and make your own beer.