The first number (eg. 8x) is the magnification power. As the magnification increases the size of the subject (eg. bird) increases. 10x power magnification helps you see a subject that is further away.
The second number (eg. 42) is a measurement (in mm) of the diameter of the objective glass (aka the front piece opposite your eyes). As the size of the objective piece increases, the binoculars will gather more light which is better in dimmer conditions.
The most common choice for birding is 8×42 (“Standard” in the above table).
Why not just get a pair of super high power binoculars with huge objective lens?
Like anything in physics, there are trade-offs. As magnification increases, the field of view becomes narrower, leading to difficulty in image stabilization making it harder to follow little birds in flight. Also, as the magnification increases, the binoculars will gather less light and the field becomes darker, making it harder to distinguish colours.
As for the objective size, as it increases the binoculars will tend to get heavy and harder to hand hold. (Glass is actually quite heavy). The cost also tends to climb significantly with larger objectives.
All of this means that 8×42 is usually the optimum for bird watching. But…. there’s always a “but”.
Going to 10x power is fine too and is somewhat better for longer distance birds like shorebirds.
Larger objectives can be good for use in low light, so long as they don’t get too heavy.
Objectives of 21 or 25 are considered “Compact” and good for small hands, perfect for travel and concerts/events, but quality glass is again expensive.
*A special note on the Pentax Papilio II 8.5 x 21: This optic has a close focus of 0.5m, making it perfect for botanists & bug lovers.
The approach we usually suggest is to start by testing an 8×42 in your budget, and then adjust size and magnification (if need be) to best suit your specific needs. We will be more than happy to walk you through all the choices so that you get the BEST optic for YOU.