A barlow multiplies the magnification of every eyepiece you own, and the choice of multiplier decides whether it is a useful addition to the case or a piece of glass that sits in it unused for years.
Which telescope do I need? Ask us below
Why 2x is the usual answer
A 2x barlow turns two eyepieces into four magnifications spread sensibly across the range. It also keeps the eye relief of the longer eyepiece while delivering the magnification of a shorter one, which is a real comfort gain at high power and matters a great deal if you wear glasses at the eyepiece.
When a barlow lens 3x makes sense, and when it does not
A 3x is useful in a short focal length instrument where even a short eyepiece does not reach high magnification. In a long instrument it usually pushes past the useful limit: a 3x on a 1200mm telescope with a 10mm eyepiece gives 360x, which is beyond what almost any night's air supports.
What to look for in the glass
A cheap barlow degrades every eyepiece used with it, which defeats the purpose. Two element barlows are adequate at long focal ratios; three element apochromatic ones hold up in fast instruments. A barlow bundled free with a budget telescope is usually the first thing worth replacing.
Questions people ask about barlow lens
Should I buy a 2x or 3x barlow?
Two times for almost everyone. Three times only in a short focal length instrument, and even then it will often exceed what the air allows.
Do telescope barlow lenses reduce image quality?
A good one is essentially invisible. A poor one softens the image noticeably, which is why the bundled barlows in cheap packages have a bad reputation.
Can I use a barlow with any eyepiece?
Yes, in the same barrel size. The barlow goes into the focuser first and the eyepiece into the barlow.