double precision
When people say a calculation is done 'in double', they mean it uses 64-bit floating-point numbers, the workhorse format of scientific computing. It is called double because it uses twice the storage of the older 32-bit single precision, and that extra room buys both more significant digits and a far wider range. It is the default floating-point type in most languages and the format almost all numerical libraries assume.
An IEEE 754 binary64 number packs 1 sign bit, 11 exponent bits (bias 1023), and 52 stored fraction bits. With the implicit leading 1 that is 53 significand bits, giving about 15 to 17 significant decimal digits. The exponent range lets it represent magnitudes from roughly 2.2 * 10^(-308) up to about 1.8 * 10^308 (with subnormals reaching smaller still). Its unit roundoff is u = 2^(-53) ~ 1.1 * 10^(-16), so a single rounded operation is correct to about 16 digits. By contrast single precision (binary32) carries only ~7 digits with u = 2^(-24) ~ 6 * 10^(-8).
Double is the comfortable default because 16 digits leaves generous slack for rounding error to accumulate before it corrupts an answer you care to ~6 digits. But it is not magic: an ill-conditioned problem with condition number near 10^8 still loses about 8 of those 16 digits no matter how stable your code, and a long enough or careless computation can exhaust the rest. Meanwhile single (and half, FP16/bfloat16) precision is making a comeback in machine learning and on GPUs, where speed and memory bandwidth matter more than the last few digits.
Storing one billion and one, 1000000001.0, in double is exact and leaves room to spare; in single precision it rounds to 1000000000.0 because single's 24 significand bits cannot resolve a unit step at that magnitude (the gap there is 64). This is why summing many values, or working with timestamps in seconds, usually demands double.
Double's ~16 digits versus single's ~7 — the magnitude where it bites.
Double precision is not 'twice as precise' as single; it has more than double the significand bits (53 vs 24), so roughly 9 extra decimal digits — and a vastly larger exponent range, not just finer steps.