To treat (or handle) with “kid gloves” is a popular American phrase meaning to treat a someone or something with extreme tact or gentleness. Sometimes mis-quoted as “kit gloves”, the term “kid gloves” originally referred to gloves that were made from the skin of a young goat or lamb. These gloves were softer and finer than gloves made from other leathers and were a symbol of gentility and elegance in the 1800′s.
The Oxford English Dictionary sites this phrase as first being used in the 1830′s. In the 1850s, saying that someone “wore kid gloves” was another way to say that a person was dainty or someone who avoided any exertion or everyday work. (The Straight Dope)