There are three places to accumulate latency, the input from the instrument to the computer, the processing of filters, the output to the monitor (headphones/speakers)/recording. Sometimes you can get away with 2 or 3 ms of latency, but anything over 5 ms is super frustrating. Remember, you're fighting the latency between plucking a string or hitting a key and when the computer acknowledging the data and sending it back out to the monitor which you're using as your guide. Best case you go and "massage" the new track to line up with the existing tracks, worst case it sounds like an out of sync high school marching band.

EDIT: The concerns here are primarily with input latency. Between plucking a string and heading it in your monitor it has to go through: your input hardware, the USB interface, the OS, the browser (which doesn't have explicit low latency capabilities), and JS. Most platforms support ASIO which is a low-level driver for reading audio data from devices. About as close to reading the ADCs yourself. Without a low-latency driver working with the OS there's so much latency overhead it's audible.

Because when you want to record your instrument along with whatever else is in your project, timing is critical and everything needs to line up.

You cannot be performing to audio that you are hearing with any delay, especially if the monitoring of the live audio is also being routed through software.

At a certain point of latency it introduced delays and badly affects how you perform. In some circumstances it makes performance actually impossible.

There are ways around this, namely if the software knows exactly what the input and output latency is then the playback and recording can be compensated. For live monitoring though you really need that done in the audio hardware itself in hard real time.

Music is very latency sensitive. If you are recording any source you generally want to have overall latency < 5ms. Input and monitoring latency is usually either handled by using fancy DSP systems or a "hack" where input audio bypasses any internal processing and gets routed directly back for monitoring.

I think the other people got a little too technical.

The reasons are things like, if you want to play in time with a previously recorded track, or if you are using digital effects and need to be able to hear their effect on your instrument as you play it.

Both of these are less of an issue today than they were 15 years ago where a USB 2.0 audio interface added significant delay into audio and made it harder to get what you wanted out of the system.

It is important if you are driving hardware synths and samplers through midi.

Otherwise latency is not really a problem.