These are various forms of "Zhuan Shu", the first style of characters to be widely used in China around 221 B.C.

This is a square form of "Zhuan Shu", which is often used for signature stamps or seals.
This script has been in use for thousands of years in China. It's also been used for seals in Japan, Korea, and elsewhere for the past 1500 years or so.

These "stamped-style" characters are between "Zhuan Shu" and "Li Shu".
These are the impressions typically left by carved stamps made from stone (sometimes called "chops").

Just after "Zhuan Shu" came into existence, this easier-to-write style called "Li Shu" was adopted. Sometimes this style is called "Official Script", "Prison Script" or "Clerical Script" in English.
It was used by the Emperor's scribes for all legal documents. The "Prison Script" title comes from the fact that thousands of copies of important declarations were hand-copied by prisoners under the order of the Emperor's court (because no printing presses existed).

This is "Kai Shu", the most widely-used form of Chinese Hanzi, Japanese Kanji, and Korean Hanja. The form shown here is the simple printed version.
This is the style that is used to teach school children to write in Asia.
Most Chinese, Japanese, and Korean people will tell you that this style is very basic, and does not contain the true art of the characters/calligraphy.
However, most tattoos in the west seem to use this style (which makes Asian people think we are weird, as this is like getting a tattoo in "Times New Roman" font in English).

These are various written and calligraphic forms of "Kai Shu".
These have a bit more artistic styling compared to the plain printed "Kai Shu".

This is a flowing version of "Kai Shu" known as "Xing-Kai Shu". The "Xing" means "grass", as in "flowing like grass". This is a slightly cursive style of writing that allows the brush to trail after each stroke so that they are almost connected.

This is "Xing Shu". This is often called "cursive" or "grass script" in English. This style of writing does not allow for the brush to be lifted very often; Thus the strokes blend together in an almost-continuous trail.

This is often referred to in English as "Ming" style after the "Ming Dynasty" when it became popular and widely used.
This script is actually the result of the first printing press developed in China during the Song Dynasty.
Because each block in the printing press had to be hand-carved from wood, this style was developed to be easier to carve and to flow with the grain of the wood.
Later, during the Ming Dynasty, printing press technology and this style of characters made it across the sea to Japan. Because of the date it was used in Japan, this style was labled "Mintai" in Japan which is "Mingti" in Chinese. However, in China this is known as "Songti". Note: "ti" = style or form.

This is "Cao Shu". This is sometimes called "chaos-style" or some variation of cursive in English. This style is under-appreciated by western audiences and is an acquired taste, even for native Asian people. Some love it, some hate it. Even native Chinese can not read it without some hints or clues (unless they are calligraphy experts).

These are variations of Chinese Hanzi / Japanese Kanji / Korean Hanja that are used for advertising and other special purposes.
The "Stone Carved" and "Saw Tooth" styles are popular for tattoos. The rest offer unique looks and possibilities in case you are seeking something far outside of "normal".

This constant-width or "pen style" character is a modern style that is a result of the "computer age". This is made to look a bit high-tech, and it actually computer-generated (nobody writes like this).

These are Simplified Chinese variations. They are sometimes the same as the traditional characters (as shown above), but occasionally, they are quite different.
There was an effort in the 1950's in China, and post-war Japan to simplify characters. The effort was independent between the two countries, so sometimes the simplification was the same and sometimes not. In many cases, Japan did not simplify.
This was in a effort to make the characters easier to write and improve literacy.
However, most Chinese and many Japanese feel that it "took the art, tradition, and ancient nature out of the characters". I do not recommend these for tattoos since your tattoo should be a form of art, and in the case of an Asian tattoo, a form of ancient art.
Please consult with me before choosing one of these simplified styles!