Iaijutsu and iaido are components of kenjutsu, the arts of the sword. It is specifically techniques used to draw the sword from the saya (sheath). This is also called battōdō or battōjutsu抜刀術.
The suffix of jutsu 術 means techniques or the art of. Batsu or nuki means to pull. Battōjutsu simply means the art of pulling out the sword.
Iaijutsu, on the other hand, has a different connotation than battōjutsu. Iai 居合 is made up of two characters, the first I 居. This character means to be or exist and also has the wider meaning of sitting. The second character, 合 means to meet or unite or join.
There is no direct translation into English of Iai. The art of using the sword in everyday situations is probably the best way to think of it. Iaijutsu does not mean solely the seated forms. There is also standing which is called tachi-iaijutsu 立居合術.
The creator of iaijutsu cannot be claimed by any one person. Although bushi such as Iizasa Choisai Ienao of Tenshin Shōden Katori Sintō-ryū (14th Century) and Hayashizaki Jinsuke Shigenobu of Shin Musō Hayashizaki Ryū (16th Century) greatly contributed to its spread and popularity.
Iaijutsu and iaidō are done from various postures from seated to standing. Although the character 居 has the older meaning of sitting, iaijutsu contains the feeling of being at "rest" in a non-battlefield situation and is more relating to that of everyday life. Sitting in a meeting, or having a meal or being served tea, etc. etc. Walking down the street is a part of this "existing in everyday life" idea.
Iaijutsu is concerned with the art of using the sword from its rested position in the saya. The person studying iaijutsu learns to deploy the sword for somewhat defensive purposes. In Japanese swordsmanship, these defensive techniques also include acts of getting the sword out to attack as well. Iaijutsu is not purely a defensive action. Techniques for stealthy or covert attacks are also included in many systems of iaijutsu and iaido.