Butt joints
This is a simple case of pushing one piece of timber squarely against the other. This is what you are doing when you use screws or nails. You could use either of these on their own, but they are stronger and stiffer when used in conjunction with PVA adhesive. Of course, you might want to be able to disassemble the piece in the future, therefore screws alone are necessary.
Nails come in various guises. Round wire nails are the norm for joinery, lost head or oval nails are for the higher quality finish needed in cabinet making. Blunt the nail head with the hammer before driving. This reduces the likelihood of splitting the wood. Keep away from the ends and edges, as this greatly increases the possibility of splits. Knock the heads of lost head and oval nails below the surface with a punch and fill with woodfiller of matching colour.
Screws are stronger than nails. Choose countersunk heads. Drill a pilot hole in the top piece of timber slightly bigger than the shank of the screw to allow it to enter the hole easily. Drill a countersink hole on top of this with a countersink bit or larger straight bit. Align this screwhole with the piece to be joined and mark through it with a bradawl. Drill a pilot hole equivalent in diameter to the screw shank to a depth slightly below that of its length. Driving in screws without predrilling is possible but often causes damage to the screwhead. Fill the countersink holes when finished.

Housing joint
This joint is suitable for joining cabinet corners and just about any vertical to horizontal component, such as fence posts and rails. A housing is a groove running across the grain. The simplest form of this is a through housing. It goes all the way through from one edge of the timber to the other. Take the piece that is to be joined and place it either edge or end on, whichever is appropriate, in the correct position where the joint is to be cut. Align at 90^o by using an engineer's square. Draw a line either side of it. These two lines represent the perimeter of the housing. Saw down to the required depth, approximately one third the total thickness of the board, on the inside of these two lines. In thicker timber, such as fence posts, it is reasonable to increase the depth of the housing to half the total thickness.
Sawing can be done by hand with a tenon saw, but the easy way to cut a housing is with a circular saw, set to the desired depth. Remember that the depth of a cut joint needs to be added to the dimensions when marking the length of your boards. Either way, cutting accuracy can be improved by clamping a piece of scrap straight edge timber across the width of the board to use as a parallel guide. Always place blocks between the work piece and clamps to avoid damage from the clamp pressure.
Once the cutting is done the waste material should be removed using a chisel. Choose one that is equivalent or slightly less than the width of the housing. Begin by holding it bevel side down, moving from the outside to middle of the joint to avoid breakout of the timber at the edges, which is very easily done. Give yourself extra inital power by gently hitting the top of the chisel handle with a mallet. Holding the chisel upside down avoids the tendency of the blade to dig in, causing the blade to sink below the required depth of the joint. Do the same from the opposite edge. The entire process of removing the waste material is made easier by making a number of saw kerfs (cuts with the saw blade) through the joint, if you have a circular saw. Once the majority of waste is removed, turn the blade bevel side up, gently finishing it to a smooth, even, flat surface with just the power of your hands.
A through joint is clearly visible in the finished work, and if it isn't done very cleanly and accurately, you might consider the results to be disappointing. The solution is a stopped joint. This simply means that you don't go all the way through, so you can't see the housing. However, the tool required to do this job easily and effectively is a router, because it is difficult to saw without going from one edge to the other. Mark out the joint in the same way and set up a parallel guide, but this time remove the waste with the router set to the required depth.
For housings at the end of a workpiece, such as cabinet corners, you have two options for modifying the joint. The simplest is to cut the housing at the very end in the normal way, turning into a lap joint, because it only has one side to the groove, not the usual two. The more sophisticated way is to cut the joint to one half of its actual width, leaving some of the material remaining at the end of the board. This leaves you a true housing, but the piece to be joined with it must have a matching shoulder cut into it, half the thickness of the board, which is equal to that of the housing. Such a joint must be cut to allow an easy fit, because trying to force the two pieces together can easily breakout the small strip of end material.
Before assembly, housings joints should be covered with a thin even layer of PVA adhesive. Once assembled, the joint should be held in place with sash or T bar clamps whilst the glue dries. There really is no substitiute for having these clamps for such work, and here lies a clear demonstration for one of the benefits of using screw butt joints, as they pull the work pieces together without needing clamps.

Tongue and groove (as a substitute for mortise and tenon)
You are probably familiar with tongue and groove as a way of edge jointing (see below) planks of wood, typically as floorboards. However, I use it as an alternative to the mortise and tenon for most lightweight tasks such as cabinet doors. Essentially, the tongue is a small scale tenon, similarly the groove is a mortise. You will require a router to be able to do this, or a plough/multi plane, but the latter is a rarity today.
Along the vertical members (stiles) you need to route a groove across its entire length. Here I assume that the groove will not only be taking the tongue with which the rail will be joined to it, but also the panel that would be part of a cabinet door. Such a panel would be either plywood or solid wood. The width of the groove should be no more than one third of the thickness of the workpiece, but you can make slight adjustment of this to conveniently fit the panel.
Set up your parallel guide fence on your router to make the groove dead centre. It is difficult use a router on the edge of a thin workpiece, and the easiest solution is to use two or three lengths side by side to give the router a firmer base. Set the required depth, 6mm is suitable, and cut the groove using at least three passes, because this reduces the strain on the router and cutter.
The next step is to cut a tongue into the end of the rail to match the groove. The tongue should generally be one third the thickness of the rail, but most importantly matching the width and depth of the groove and positioned dead centre. Mark out where the waste material should be removed. You can use a rebate cutter with a guide bearing, simply setting the correct depth of cut. Otherwise you can use a straight cutter, setting up a guide fence across the width of the workpiece with a piece of straight edge timber and G clamps. You can check your joints before assembly with a dry run, including the panel. When you are ready, cover the tongues in an even thin layer of PVA adhesive. Push all the components together and clamp from stile to stile across the rail with sash clamps. Leave overnight to set.

Edge joints
It is fairly uncommon to come across very wide planks of wood. If you can find them, they are highly prone to cupping when they are plain sawn, which most are. Therefore, you create wide boards from gluing smaller components together. The good news is that this is quite easy.
Firstly, no cutting of joints is required. You simply glue one straight edge to the other. It is by nature very strong. Tongue and groove joints are not required. You see these in floorboards and matchboarding, but they simply provide a way of joining two pieces without gluing but allowing for any expansion to avoid the appearance of gaps.
The requirement is that all the planks that you want to join must be planed true with flat sides and square edges. TImber, even ready planed, is often cupped. The only way around this is to plane it flat again. Without this prerequisite, an accurate joint will not be obtained. Cover the edges to be joined with an even coating of PVA adhesive. Place the boards together edge to edge with the annual rings of the end grain in opposite directions. Clamp them together with sash clamps, making sure the surfaces of each piece is level with the other.