When I enter $-145 \mod 63$ into google and some other calculators, I get $44$. But when I try to calculate it by hand I get that $-145/63$ is $-2$ with a remainder of $-19$. This makes sense to me, because $63\cdot (-2) = -126$, and $-126 - 19 = -145$.
So why do the calculators give that the answer is $44$?
I think you have to start with the more basic question, "What does $\text{mod}$ mean?"
When we say "$\pmod{63}$" what we really mean is: Pretend that the "number line" is bent around in a circle so that when counting up, after you hit $62$ you return to $0$. On such a "number circle", the numbers $5,68, 131, 194, \dots$ are all equal to each other. And you can count downwards, too: $68, 5, -58, -121, \dots$ are also all equal.
It's common to interpret $a \pmod{63}$ to mean "Find the number between $0$ and $62$ that is equal to $a$, mod $63$." You can always find such a number by repeatedly adding or subtracting 63 to your given number until you get it into the desired range.
In this case, $-145 = -82 = -19 = 44 = 107 = \dots$. The only result that lies between $0$ and $62$ is $44$.
Note, though, that you are not wrong in thinking that $-145 \pmod{63} = -19$. When working mod $63$, the numbers $-19$ and $44$ are identical.
Positive $145$ divided by $63$ is $2$ with a remainder of $19$, since $145=(2)63+19$.
However, $-145$ divided by $63$ is $-3$ with a remainder of $44$, since $-145=(-3)63+44$.
Remainders need to be positive. When dividing by $63$, they are between $0$ and $62$ inclusive.
We say $a \equiv b$ (mod n) if $a-b$ is a multiple of $n$. So notice that:
$$-145-44 = -189 = -3(63)$$
a=b (mod c) iff c|(a-b)
In your case 63|(-145-44)
Some calculations using modular arithmetic: $$-145\equiv_{63}-145+63\cdot3\equiv_{63}-145+189\equiv_{63}44\equiv_{63}44-63\equiv_{63}-19$$
-19 = -145 mod 63.
44 = -145 mod 63.
107 = - 145 mod 63.
56550671 = -145 mod 63.
There are an infinite number of correct answers. Any number of the form: n = 63k - 145 will be a valid answer.
So why does your calculator choose 44 rather than -19 or 5650671? Probably because it was programmed to find the smallest non-negative value. Probably. Some programs are programmed differently and would give -19.
%in the C language family) handle mod of negatives in different ways, by the way, but in my experience as a programmer I almost always want the positive result, i.e. 44 instead of -19. – Russell Borogove Dec 17 '15 at 21:18