行書 & 草書 (semi-cursive and cursive writings)
行書【ぎょうしょ】 (semi-cursive script) is similar to English 'handwriting' style, and this is the most orthodox way of writing Japanese sentences fast. This is what Japanese students learn at middle school, although that does not necessarily mean all students master beautiful 行書. You can compare 楷書【かいしょ】 (regular style) and 行書 in this page.
草書【そうしょ】 (cursive script) refers to more aggressively abbreviated kanji/kana. Ordinary Japanese people do not know how to write these, and have difficulty reading 草書.
略字 (abbreviated character)
While 行書 and 草書 are standardized and sophisticated, people often simplify kanji in non-standard ways. You can see the two best-known examples here. To take an extreme example, 慶応【けいおう】 is sometimes written like this using English alphabets. Although being useful, these types of 略字 are generally frowned upon in formal situations, and I prefer to use 行書 whenever possible.
略語 (abbreviated words)
Similar to English acronyms. For example, 国際連合 (Unitated Nations) is shortened into 国連【こくれん】, 高等学校 (high school) is shortened into 高校【こうこう】, リモートコントローラー (remote controller) is shortened into リモコン, etc. Coining a new 略語 is not what people do very often.
In Japanese, you can't shorten long words by piking their consonants (i.e. somebody => smb, statement => stmt, ... is there a name for this?). But because kanji are ideographic characters, many things can be expressed using only one or two kanji. "Somebody" can be written as simply as "人" in a grammar textbook, etc.
There is a Japanese version of shorthand, which allows us to write Japanese as quickly as people speak Japanese. Of course this is definitely for those who are professionally trained. Example.