A reference frame is simply a system of co-ordinates measured relative to a specific point, which is the origin in that reference frame.
Often we use Cartesian co-ordinates in each reference frame (we don't have to, but this makes it simpler to define what we mean by a "straight line") and we rotate the co-ordinates in each reference frame so that the $x,y,z$ axes are aligned (again, we don't have to, but it makes life simpler). And we choose the origin in each reference frame so that all of the origins coincide at some specific time, which we call $t=0$.
We can then identify a particular point (or event) in spacetime by its co-ordinates and time relative to reference frame $A$ - say $(x_A, y_A, z_A, t)$. In another reference frame $B$ the same event will have different co-ordinates $(x_B, y_B, z_B, t)$. Note that because we are considering Newtonian mechanics here, the value of the time co-ordinate $t$ is the same in all reference frames - there is a universal time. If we were considering relativistic mechanics then $t$ would depend on the reference frame as well.
We can track the $(x_A, y_A, z_A)$ co-ordinates of some object $O$ in reference $A$ - in general these will depend on time $t$. If the $(x_A, y_A, z_A)$ co-ordinates of $O$ are constant (i.e. do not depend on $t$) then we say that $O$ is at rest relative to reference frame $A$. If the $(x_A, y_A, z_A)$ co-ordinates of $O$ depend linearly on time $t$ (so if $x_A(t) = x_A(0) + vt$ etc. ) then we say that $O$ is moving at a constant velocity relative to frame $A$.
By observing the co-ordinates of different events in reference frames $A$ and $B$, we can deduce a set of relations between the two sets of co-ordinates, and these relations hold for all events in spacetime. For example, if frame $B$ is moving relative to frame $A$ with constant velocity $v$ parallel to the $x$ axis then
$x_A = x_B + vt \\ y_A = y_B \\ z_A = z_B$
This is called a Galilean transformation. But if frame $B$ is accelerating relative to frame $A$ with constant acceleration $a$ parallel to the $x$ axis then
$x_A = x_B + \frac 1 2 at^2 \\ y_A = y_B \\ z_A = z_B$
and this is no longer a Galilean transformation.
If we have an object $O$ with no forces acting on it then we can define a reference frame $F_O$ in which this object is at rest (simply define the origin of the reference frame to be wherever that object is). Newtons' first law then says that any other object on which no forces act will either be at rest or will move with a constant velocity relative to reference frame $F_O$. And this will also be true in any other reference frame that is related to $F_O$ by a Galilean transformation.
However, Newton's first law will not be true in a reference frame that is related to $F_O$ by a non-Galilean transformation. In a reference frame that is accelerating relative to $F_O$ for example, then $O$ will appear to be accelerating even though there are no forces acting on it.