There are two major types of fabric content - natural fiber and man-made.
Like the name implies, natural fibers come from materials found in nature.
Wool - the fur from sheep, goats, rabbits, camels, etc.
Cotton - the seed pods of the cotton plant
Linen - the flax plant
Silk - the cocoon of certain moths
The fibers are combed (aligned) and spun (twisted) together. The resulting yarn can then be turned into fabric.
The yarn may be dyed before it is turned into fabric. This is called yarn-dying. This is the most common.
Alternatively, the fabric may be turned into a garment and the entire garment dyed. This is called garment-dying. You typically see this on specialty T-shirts.
One of the big advantages of natural fibers are their breathability (air and/or moisture can easily flow through them.)
Man-made fibers are typically made of petroleum products. These fabrics include nylon, polyester, lycra, Dacron, etc.
The material starts as a molten liquid and the color is added before the fiber is extruded. The resulting fibers may or may not be spun.
Since the color is integral to the fiber, you tend not to be able to dye man-made fabrics successfully. There are some specialty dyes for man-made fibers, but the color will only sit on the surface of the fiber and may not wear well.
Man-made fibers tend to wear well (not wear out quickly) and they tend not to wrinkle as much, however they do not breathe well.
You can find blends of different natural fibers (like a silk/wool blend) or natural fibers mixed with man-made fibers (like a poly blend). The end of the fabric bolt should indicate the contents of the fabric. If you buy a ready made garment, the garment tag should contain the fabric content.
Blends of all natural fibers can usually be dyed. You may not get the results you expect if you try to dye a natural/man-made blend. The natural part of the fabric will accept the dye while the man-made part will not. Typically you will end up with a pastel color.
There are three major ways that fabric is made - woven, knit, and bonded or nonwoven.
As a kid, you may have made your mom a potholder out of fabric loops or a mat made of woven construction paper. These are simple versions of weaving. In fabric, horizontal threads are interlaced over and under vertical threads. The number of vertical threads passed over or under gives your different types of weaves - plain, satin, or twill.
Woven fabrics tend to be very stable and non-stretchy along the vertical and horizontal threads. This is why your jeans get so uncomfortable after chowing down at a holiday meal.
Knit fabrics are made of interlocking loops of yarn or thread. There are different knitting stitches that create different types of patterns
Knit fabrics have more elasticity, or stretchiness. The amount of stretch depends on the type of knit. Sewing patterns designed for knit fabrics often have a guage on the backside to determine if the fabric has enough stretch for the pattern.
The stretch can be two-way (stretches along the width OR the length) or four-way (stretches along the width and the length)
Bonded or NonWoven
In these type of fabrics, the fibers are pressed together using heat and/or moisture. Felt is probably the most common type of nonwoven fabric. These fabrics do not stretch.
Edited to add: any of these fabric types can be made of any fabric content. Example - you can have a knitted silk, or a woven polyester. Felt can be made of acrylic or wool.
Perhaps the most important concept in fabric is grain. If your pattern pieces are not cut along the true grain lines, your garment may never hang properly, develop strange wrinkles, or bulges. There are not fixes for garments that aren't cut on the grain.
There are three terms that have to do with grain.
Lengthwise grain refers to the threads in fabric which run the length of the fabric, parallel to the selvedge (the factory edge) of the fabric.
Crosswise grain are the threads that run perpendicular to the selvedge of the fabric or the cut edge of the fabric as it comes off the bolt.
Bias grain is the thread line that is at a 45 degree angle to the lengthwise and crosswise grain of the fabric as it is on the bolt. The bias has stretch in woven fabric and will hang differently than a garment that has been cut on the straight or crosswise grain.
Top to bottom of this picture is the lengthwise grain. Sewing patterns are most commonly laid out on the lengthwise grain.
Side to side of this picture is the crosswise grain, or cross grain. Unfrequently a pattern piece will call for a cross grain layout. To ensure you have a straight cross grain on the end of your fabric is to either pull a thread out and cut along the space left between the threads on either side, or to tear your fabric. It will tear along the cross grain.
Is Your Fabric On-Grain?
You know your fabric is on-grain when the lengthwise and cross grain meet at a 90 degree angle, as in this picture. Look closely at your fabric and if the threads do not meet at a 90 degree angle, you can fix that by pulling the fabric on the bias.
Bias grain is used when you want to change the drape of the fabric. Clingy red carpet gowns or on silent films stars are cut on the bias. Warning - bias pattern layouts require a lot of fabric.
What About Grain Lines on Knits?
Grainlines apply to both woven and knit fabrics. The stretchiness of the knit may vary depending on whether it is the lengthwise or cross grain. The back of your pattern envelope will tell you how much stretch is required and the direction required.
Areas to Watch
Many fabrics are sold folded in half and wrapped around a flat-ish cardboard core. One of the biggest mistakes I see novice sewers make is to assume the the fabric is folded on the grain. While the machines that wind the fabric on the bolts are better than they were in the 60s, they still aren't perfect.