For an elliptic curve, what is the difference between the base field modulus $Q$ and subgroup $r$ What is the difference between the basefield modulus $Q$ and a subgroup of prime order $r$?

They are all fields, but what is their relevance to the curve they are defined upon?

How does this relate to a scalar field?

Also: Which field is the curve defined upon?
You are asking about a subset of Elliptic Curve (EC) domain parameters over $\mathbb{F}_p$ or $\mathbb{F}_{2^m}$

Domain parameters of Elliptic Curves over $\mathbb{F}_p$ is the sextuple:

$$T= (p,a,b,G,n,h)$$


The integer $p$ defines the field $\mathbb{F}_p$,
$a$ and $b$ defines the curve equation in Weierstrass form: 
$$E: y^2 \equiv x^3 + a \cdot x + b \pmod{p}$$
$G=(x_g,y_g)$ defines the base point
a prime $n$ which is order of $G$
and an integer $h = \#E(\mathbb{F}_p)/n$


So, first choose an EC, then we choose a $G$ that generates a subgroup of order $n$.


  They are all fields, but what is their relevance to the curve they are defined upon?


EC is an Algebraic Group but it is defined over the Field $\mathbb{F}_p$


  How does this relate to a scalar field?


We have scalar (point) multiplication to represent the $k$ addition of a point $P$ by itself as $kP$.


  Which field is the curve defined upon?


Above $\mathbb{F}_p$, it can be also $\mathbb{F}_{2^m}$, but these are for general Cryptographic purposes. EC's can be defined over the Complex, too.