$$q\in T \text{ is a minimum element of }T\text{ means }(\forall x\in T\setminus\{q\}: q R x\\ q\in T \text{ is a minimal element of }T\text{ means }(\forall y\in T\setminus\{q\}): y !R q.$$
Those are the definition in my book. I don't understand how for the minimum, there could be an order relation and then for minimal, there is no order relation? What are some examples that could illustrate that (would be helpful using integers or regular numbers)?
Can someone help me understand that and how I can prove that if q is a minimum, then q is a minimal using the 2 definitions listed?
You go about proving like you go about proving any logical statement.
What you know is:
$$\forall x\in T\setminus\{q\}: qRx.$$
You want to prove:
$$\forall y\in T\setminus\{q\}: y!Rq.$$
To do that, you take an arbitrary $y\in T\setminus \{q\}$. Now, for this element $y$, you need to prove that $y!R q$. For this, you can use the fact that $y\in T\setminus \{q\}$, therefore, you know that $qRx$. You also probably know some properties of $R$ which will help you in your proof.