A barlow is a small negative lens that sits between the eyepiece and the telescope and multiplies the instrument's effective focal length, so a 2x barlow doubles the magnification of every eyepiece you own. That is why it is the most recommended second accessory on this shelf.
What it actually multiplies
Magnification is the telescope's focal length divided by the eyepiece's, so a 300mm instrument with a 20mm eyepiece gives fifteen times. Put a 2x barlow in front of that eyepiece and the telescope behaves as though it were 600mm, so the same eyepiece gives thirty times. Nothing about the light gathered has changed: the same photons now cover a larger, dimmer image.
When it earns its place
A barlow doubles the useful range of a small eyepiece set, which is the honest case for buying one. Two eyepieces plus a barlow give four magnifications, and the barlow costs less than a third eyepiece of comparable quality. It also lets you keep the long eye relief of a low power eyepiece at high power, which matters if you wear glasses at the eyepiece.
When it does nothing for you
Every telescope has a magnification above which the image gets bigger and worse, set by its aperture and by the steadiness of the air. On a 76mm instrument that ceiling is around 150 times on a good night. If a 2x barlow pushes you past it, what you gain is a larger blur, and the fault is not the barlow's.
The quality question
A cheap barlow degrades every eyepiece used with it, which defeats the point of owning one. The barlows bundled with budget telescope packages are usually the weakest component in the box, and replacing one with a decent two or three element unit is among the cheapest real improvements available.
Questions people ask about barlow lens
What does a barlow lens do?
It multiplies the telescope's effective focal length, usually by two or three, which multiplies the magnification of whatever eyepiece is used with it. It adds no light and no resolution.
How do I use a barlow lens?
The barlow goes into the focuser or diagonal first, then the eyepiece goes into the barlow. Refocus afterwards, because the focus point moves noticeably.
Is a 2x or a 3x barlow better?
A 2x for most instruments. A 3x pushes past the usable magnification of anything under about 150mm of aperture on most nights.