A "Hot" wire carries the electricity from the power plant into the house. The "neutral" wire carries the electricity out of the house. Because that was the minimum required, a whole lot of houses were built with just those wires.
(There's usually 2 hot wires and 1 neutral for reasons we don't need to go into right now.)
As long as there is no electrical fault with the devices plugged into your electric box, all the current will flow from the hot, through any devices that are turned on, and back out the neutral. Problem is, electrical faults can develop. The fault can manage to connect the electricity to the outside of the device. Touching it results in a nasty shock. Enter grounding.
Newer houses still have those two wires bringing the power into and out of the house. But there's a 3rd wire embedded in the ground. There's either a long rod pounded into the earth, or the wire is attached to the steel in the foundation, and thus embedded in the ground. In electrical devices with 3-prong plugs, the ground plug is attached to the outside of the device. So if an electrical fault develops and energizes the wrong parts of the device, the current will flow out the ground wire. This makes the devices significantly safer. It also provides a path for surge protectors to discharge any surges. If your surge protector isn't plugged into a grounded outlet, it won't be able to do anything.
So are two-prong devices inherently dangerous? No. Today they're "double-insulated", meaning an electrical fault wouldn't be able to energize the outside of the device.
What's GFCI? (The plugs in the bathroom and kitchen with the buttons in them) A GFCI (Ground Fault Circuit Interrupter) is a major improvement over basic grounding. The plug (or circuit breaker) monitors the current flowing out on the hot and back in on the neutral. If the measurements are not the same, it disconnects the circuit. If you throw a grounded hairdryer into the bathtub, the water and plumbing is too good of a conductor - current will still flow through the water and anyone in that bathtub. So a GFCI outlet/circuit is installed in wet locations to protect against that danger. You'll note that GFCI doesn't actually require a ground - since the plug is monitoring the hot and neutral, a ground isn't required for the GFCI to work.
Long story short: Lack of grounding plus poorly maintained/old electrical devices can be a safety issue, which is why grounding became the norm. It is unlikely that you will suffer any ill effects from lack of grounding (but it is possible). Any electronics you want to protect with a surge protector needs to be plugged into a grounded outlet.
What to do about it:
Option 1: Nothing. Hasn't killed you yet, and while injury is possible, it is statistically unlikely.
Option 2: Have an electrician rewire the house. Your plugs and switches only have two wires. A 3rd wire needs to be run in order to ground them. This gets pretty expensive.
Option 3: If there are certain outlets where you want grounding (Washing machine, electric dryer, computer/other electronics) grounding a single outlet is fairly easy. You typically run a wire from the plug to a plumbing pipe, since those are usually metal and embedded in the ground.
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