These are various forms of seal script, the first style to be widely-used in China around 221 B.C.
In Chinese, this is known as "Zhuanshu". Sometimes referred to simply as Zhuan, or Xiao Zhuan (xiao = small -> "small seal" as opposed to the larger square style). In Japanese, Zhuan is pronounced "Ten" hense "Tensho" in Japanese.

This is a square form of seal script, 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.
Romanized as "Fang-Zhuanshu" from Chinese. It's just "Tensho" in Japanese.

These "stamped-style" characters are between seal script and official script.

These are the impressions typically left by carved stamps made from stone (sometimes called "chops").
Often referred to as "Gu Zhuan" or "stamp seal" in Chinese.

Just after seal script came into existence,this easier-to-write style called official script was adopted. Sometimes this style is also called "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).
Romanized as Lishu from Chinese and Reisho or sometimes Leisho from Japanese.

These are printed forms of regular script. They're the most widely-used general styles of Chinese Hanzi, Japanese Kanji, and Korean Hanja.
Kaishuprint 1-5 are typical of the styles used to teach school children to write in Asia.
Most Chinese, Japanese, and Korean people will tell you that in printed form, these look very basic, and do not contain the true art of the characters/calligraphy.
However, most tattoos in the west seem to use this kind of printed style. Using a this is the equivalent of getting a tattoo in "Arial" font in English.
Romanized as Kaishu from Chinese and Kaisho from Japanese.

I recently added Kaishuprint 5-11. I recommend Kaishuprint7 and Kaishuprint8 if you like this general printed style.

These are various written and calligraphic forms of regular script.
These have a bit more artistic styling compared to the plain printed regular script above.

This is a flowing version of regular script known as "Xing-Kaishu" from Chinese. The "Xing" means "running", "flowing", "traveling". This is a slightly cursive style of writing that allows the brush to trail after each stroke so that they are almost connected.
Romanized in a different word-order as "Kai-Gyosho" from Japanese.

This is running script. This is often referred to as "cursive" 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.
Romanized as Xingshu from Chinese or Gyosho from Japanese.

Note: The differences between this script and "running regular script" are very subtle and subjective. In some cases, we've mixed them up here.

This is sometimes called "chaos-style", "grass script", 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).
Romanized as Caoshu or Tsaoshu from Chinese and Sosho from Japanese. The literal meaning from Chinese and Japanese is "grass script".

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" or "Mincho" in Japan ("Mintai" would be "Mingti" in Chinese). However, in China this is known as "Songti". Note: "ti" = style or form.

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).