Watts are a measure of electrical power used. It's not a measure of light given off. Unfortunately most people learned that a 100 Watt light bulb is so bright, and a 60 Watt light bulb is so bright, and so on. So they've got a little table in their heads which measure brightness in Watts, even though it's the wrong unit.
When newer lighting tech like fluorescent (CFL) and LED became popular, this became a problem. Because they're more efficient, they give off more light per Watt than an incandescent bulb. Or put another way, they use fewer Watts to produce the same amount of light.
Equivalent-watts were made up to bridge this gap.
■The Watts on the label of a light bulb is always how much power it uses. People in forums and blogs might use the term incorrectly, but the manufacturers know what a Watt is and will always label it correctly.
■The equivalent-watts on the label is how much light it gives off compared to an incandescent bulb.
So Watts are the power used, equivalent-watts are the brightness. So in your example of a 150 equivalent-watt bulb which is rated for 60 Watts, it consumes only 60 Watts but gives off as much light as a 150 Watt incandescent bulb. And yes it's safe to use in a 60 Watt socket.