A Barcode Symbology defines the technical details of a particular type of barcode: the width of the bars, character set, method of encoding, checksum specifications, etc. Most users are more interested in the general capabilities of a particular symbology (how much and what kind of data can it hold, what are its common uses, etc) than in the excruciating technical details.
Without putting every barcode and every symbology, we'll go over the ones you will see most.
3of9 or CODE39
1D Alphanumeric and some symbols, usually 8-13 characters but can be more or less depending on space. Used everywhere especially in manufacturing, looks like a ladder sideways.
A 1D code similar to 3of9 but some feel this is a better selection for manufacturing, logistics, and shipping. The density is better and it offers a larger selection of charaters than 3of9. Ladder style and alphnumeric.
Used mainly in the US for all retail stores, especially grocery stores. Registration fee is $750 for one of these thru Uniform Code Council,a t (937) 435-3870. Company is defined by the first five after the lead character and the next five define the company. Numeric only. Ladder and numeric on bottom.
2D or 2 dimensional barcode that can hold a large amount of data. Typically used on drivers license. Looks like a messed up ladder barcode.
Another of the 2D barcodes. It is especially used when the spacing for the code is small. This is used in etching and and when a tiny code is needed for space restrictions. Advil bottles is one example. The UID code standard for the US government is Datamatrix. Shaped like a square in 4 quadrants.
Used by USPS to track mail and is on the envelopes we receive with our mail.
Many others exist but this is a sampling of most barcode symbology that is used today.