Ground Fault Circuit Interruptors.

About

Main points: a GFCI is not grounded; it doesn't need the circuit to have the 3rd wire. It just measures if current in 2 lines is balanced. It protect people better than a ground wire.
Is GFCI safer than a ground pin? Yes, in case a part of an appliance is not grounded. How a ground works - [1]. Also - just because there is a ground pin - it doesn't mean that it is connected or that it is connected properly.
Grounding without GFCI provides personal safety, safety for electronic gear, and fire safety. GFCI without grounding provides better personal safety than grounding, fire safety, but no safety at all for electronic gear. - [2]
Is it possible to both ground and use a GFCI? Apparently yes - second paragraph of top answer. [3]
According to NEC 2014, you are not allowed to ground a GFCI [4] -

Note that voltage surges are yet another thing - and a GFCI does not protect against those. Use a surge protector - [5]
GFCI is required by code for outdoor receptacles - [6], and for construction sites.
Ground fault protection protects heavy industrial equipment. Beyond the scope of GFCI discussion. [7]
Case where GFI does not work - is that dangerous? [8]. If you touch high voltage without being grounded - that appears safe - just like power linesmen can touch high voltage as long as they are not grounded. With this said - is an ungrounded person not protected by a GFCI? The GFCI doesn't trip - but does the person get hurt?
Note that GFCI does not protect you from line contact hazard - where you hold both hot wires in each hand. (In this case, the current is still balanced, but you're likely dead if you don't let go.) This is not a common occurence. However, GFCI protects against the most common form of electrical shock hazard, the ground-fault. [9]

More

Wall outlet - [10]
Portable plug - $9 - [11]
GFCI circuit breakers also exist, but are expensive at $40 each - [12]. $30 on Ebay - [13]
DIN Rail GFCI breaker - $5 - [14] - is this really GFCI?
Can be plug-in receptacles - [15]
Arc Fault Circuit Interruptors (AFCI) are another thing - [16]
Note that GFCI outlets sense differential current, and they are not grounded - [17]
Portable GFCI - GFCI on a power cord - [18]
A GFCI outlet is quite complex - with its own circuitry. See teardown and explanation - [19]
GFCI chip from On Semi - [20]