It's often said that "the best defence is a good offence", but this isn't always the best mantra. It's often beneficial to take some form of personal defensive kit so the first shot from an ambushing force doesn't turn Ensign Richards - or worse, Captain Johnson - into a fine red mist.
Personal shielding is a much lower-power extension of the same concept as is used to shield starships. It consists of a small module or set of such positioned at points around the body of the user (rather than larger emitters on the hull of a ship).
Where the power comes for all this depends on what race you're asking - the Federation draw power from the ample reserves of their commbadges. In most cases, personal shields do not bring with them sufficient power to project a cloaking field; an exception is the Jem'Hadar, who carry a larger power pack with them than other races. (They are able to do this due to their increased musculature relative to a human or a Trill; the Klingons could, but choose not to.) The Borg equips its individual units with shielding powered by small fusion generators, tiny antimatter or collapsar reactors, or in some cases just plain old-fashioned dilithium batteries.
When your shield fails, good old-fashioned armor is the way to go for personal passive defense. But providing sufficient protection from the high-energy, often directed-particle attacks of the 25th century isn't a job for simple para-aramid fibers stretched over chunks of glass.
No, these days armor systems come built in to most crew uniforms. Again, the civilization generating the uniform tends to have a great effect on the armor's thickness, durability, and capabilities. The Klingons, favoring close combat but needing energy-weapon resistance, favor thick plates of duranium and tightly-pulled leather. The Romulans weave threads of tritanium and brannon together, making knit mail that not only sinks off vast quantities of energy, but is wearable on a long-term basis. The Federation favors a carbon nanotube matrix for impact protection throughout the garment bolstered with dilithium to absorb nadion discharges. The Borg simply welds chunks of starships to its units. Each type of armor carries notable advantages and disadvantages, like shields... And each has also been tried and proven over centuries.