Shorthand typically refers to any method of writing that uses symbols for common phrases, words or sounds. Some methods of shorthand use abbreviations for words. In either case, someone well trained in shorthand should be able to take dictation as fast as someone else is speaking. Obviously, shorthand is generally used with the intention of transcribing it to longhand... but some people like Samuel Pepys don't ever do that.
There are looooots of different kinds of shorthand, including geometric systems, script, and geometric-script. Turns out speedwriting is just one of maaaaaany shorthand writing systems.
Non-symbol systems often supplement alphabetic characters by using punctuation marks as additional characters, giving special significance to capitalised letters, and sometimes using additional non-alphabetic symbols. Examples of such systems include Stenoscript, Stenospeed, Speedwriting, Forkner shorthand, Quickhand and Alpha Hand. However, there are some pure alphabetic systems, including Personal Shorthand, SuperWrite, Easy Script Speed Writing, and Agiliwriting, which limit their symbols to purely alphabetic characters. These have the added advantage that they can also be typed - for instance, onto a computer, PDA, or cellphone. Early editions of Speedwriting were also adapted so that they could be written on a typewriter, and therefore would possess the same advantage.
This is an example of the Lord's Prayer written in a bunch of different shorthand systems.
About speedwriting specifically:
Speedwriting is phonetic with a ‘k’ used for a hard c, ‘C’ for ‘ch’, ‘j’ for ‘g’ in ‘age’. It condenses words by omitting silent letters and only writing long vowels, and initial short vowels. Sentences are ended with ‘\’ and a ‘/’ is used for omitted syllables. There are other abbreviating devices, including capitalisation, and the use of punctuation marks to denote combinations of sounds. It uses around 100 abbreviations for common words and suffixes.
Speedwriting uses a stylized script made in 1942 for faster handwriting, in which the ‘t’ is uncrossed (l is looped to distinguish them), ‘i’ is not dotted, ‘m’ is a simple curve like a stretched ‘n’ and 'w' is also a simple curve like a stretched 'u'.
Speedwriting is more than twice as fast as longhand, due to using half the letters, but it is nowhere near as fast as symbolic shorthand systems. Speeds of up to 120 words a minute are possible for short periods of time, with speeds of 80 words a minute being regularly attained. It is therefore more useful for someone wanting a simple system to speed up handwritten note taking than for reporting.
Here's what it looks like:
I think we should all just type real fast and forget about all this abbreviated non-sense. Writing things by hand holds nothing more than aesthetic value for me. Yes, having said that, I really like my handwriting.