Specifying a resistor as "1%" is not really sufficient to describe it, but it is often used as kind of a short form. One would expect a "1%" resistor to have tolerance within +/-1% (of course)- probably quite a bit better- see below, but also to be reasonably stable with temperature (perhaps +/-100ppm/°C or better- again probably quite a bit better typically) and to not change much with humidity and when it is soldered, and with time. By default it would likely be a film type (probably metal film), so would have limited pulse-handling capability, so you can't always substitute a 1% part for a 10% part.
If you use a 10% resistor selected for 1% in a measuring circuit you may find that it is unstable after soldering, with temperature, with mechanical stress, with time or with humidity and your circuit is not as stable as intended.
The materials used for a precision resistor can be quite different from those used for a non-precision part- as well as better trimming machines and so on. By the way, at least for the past 30 years or so, resistors are made by trimming on automated machines (even 5% resistors) you'll generally find some statistical correlation between values of resistors in the same batch. Usually they'll cluster around a value that's a bit different from the nominal value, and they'll typically be within about 1/3 to 1/5 of the nominal tolerance (so a 5% resistor is usually within +/-1% to 1.5% and a 1% resistor is usually within +/-0.2% to +/-0.3% of nominal. That's a consequence of wanting to get almost all the resistors within tolerance so none have to be discarded.
For example, a quick measurement of four 8.25K 1% 0603 resistors (Rohm) gives a mean value of 8.26128 and a standard deviation of 0.01433. If those statistics were representative (too small a sample), and the distribution was Gaussian, maybe one in a million would fall outside the 1% band.