I also sort of lied, as upon reflection perhaps it is better to discuss basic leaf anatomy before going on to how the process of photosynthesis actually works. But I'll make it a long post to make up for it ;)
So there are a few basic components to leaves, that are common to all no matter what species of plant we're talking about. I'll list these, along with their function below:
Cuticle - A waxy layer that sits on the surface of the leaf, it's main role is protecting the leaf from attack from pathogens and to prevent water loss.
Epidermis - This is a single cell layer that surrounds the outside of the leaf (cuticle sits on this layer). It functions much like our skin, in the sense that it prevents nasties from getting in, and water and the insides of the leaf from escaping. There are openings called stomata within this layer, that allow gases to move in and out of the leaf. These pores are controlled by the plant, which can open and close them as it needs. Some plants also have hairs arising from their epidermis, and these have multiple functions (protect from predation, reflects excess light, help reduce water loss through the creation of cool and humid microclimates around the stomata) although which function they perform varies with species.
Vascular tissue - Much like our veins and arteries, this is made up of Xylem tissue, which conducts water and it's associated dissolved nutrients, and Phloem tissue which carries photosynthates (ie the sugary products of photosynthesis) around the plant.
Mesophyll tissue - This is where it gets interesting. Mesophyll is a tissue that is usually found in 2 forms in leaves (although to be fair some leaves only have one type).
The first is Palisade mesophyll (also known as palisade parenchyma), and is composed of elongated, block-like cells that are usually stacked along the sun-side of the leaf. These cells collect light and are full of chloroplasts. You guessed it, they're the photosythetic cells in the leaf. However CO2 is also needed for photosynthesis, and that's where the second type of mesophyll comes in.
Spongy mesophyll (or spongy parenchyma) is (surprise surprise) called such because it is full of air-spaces and so looks like a sponge. The airspaces allow gases to pass in and out of the leaf easily whilst minimizing water loss.
So those are the basic tissue types (there are sometimes others, such as fibers for structural support or other specialized cells in various different plants), however their arrangement within a leaf can vary dramatically.
Basically the aim of any leaf is to photosynthesise, whilst not losing too much water, so the tissues within the leaf will be arranged to best accomplish this in the environment that the plant grows in.
For example, a plant that grows in a hot, dry climate is more likely to have a thick cuticle and less spongy mesophyll than a plant that grows in cooler wetter environments, as it will need to conserve water whereas the cool climate plant won't.
Perhaps the best example of changing the arrangement of tissues within a leaf to suit the environment, is to compare a typical European plant, such as a privet leaf, to an Australian plant adapted for hot dry environments, such as a Eucalyptus leaf.
Pictures below (I don't have access to a camera that I can take micrographs on at the moment, so the sources were: http://sols.unlv.edu/Schulte/Anatomy/Leaves/PrivetLeaf.jpg for the privet and http://www.sciencephoto.com/media/98433/enlarge for the Eucalyptus)
Here (left) we have the typical leaf. Along the top is that palisade mesophyll, and this is the side of the leaf that would be in the sun. The leaf is held parallel to the ground (known as a dorsoventral leaf), meaning the sun always hits the upper surface of the leaf no matter what time of day it is. So to sum up it makes sense to pack that side with the light-collecting palisade mesophyll, and have the spongy on the cooler side of the leaf that is permanently shaded.
And this one (right) is of the Eucalyptus leaf. You can see there is lots of palisade mesophyll, but hardly any spongy. The big red bit in the middle is a vascular bundle - the reason it's so huge is that it's the central vein in the leaf - and the white holes are actually oil glands. But the mesophyll is what we're interested in here ;)
It's arranged like this (ie palisade is packed on both sides of the leaf) because Eucalyptus species have their leaves hanging downwards, perpendicular to the ground. This is known as an isobilateral orientation, and helps reduce waterloss and photodamage during the hot part of the day. Because the leaf hangs downwards, during the morning and afternoon (when the sun is cooler) the light hits one side of the leaf. As it moves overhead at midday and becomes hotter, there is very very little leaf exposed to the sun (sort of like a person - in the morning and afternoon the sun will strike your whole body when you're standing up, but at midday only the top of your head will be hit by the sun). And in the afternoon the other side of the leaf is in the sun. This system helps keep the leaf cool, but means that stomata and palisade mesophyll are needed on both sides of the leaf.
So that, in a nutshell, is basic leaf anatomy. There are minor differences between monocotyledon and dicotyledons (for definitions of those types of plants, see wikipedia), and of course different types of plant may have different specialisations, but for your general C3 plant, that's what it looks like. C4 are slightly different, but for a proper explanation of what the hell C3 and C4 even mean, and how they're different, you'll have to wait for the next post ;)