Newton’s second law can be written as
$$m a = f_1 + f_2 + \cdots + f_n$$
The term $m a$ having dimensions of force and being proportional to the mass of the body, which is a measure of its inertia, is often named inertia force. So, according to the previous equation, the inertia force is just the resultant force acting on the body under analysis. In Fluid Mechanics it is advantageous to use mass per unit volume of the body (fluid in this case), that is its density, so that Newton’s law (or rather, the Navier-Stokes equation) is written with the terms having dimensions of force per unit volume of fluid.
When fluids flow, different types of forces act on the fluid. These are represented in the previous equation by $f_1$, $f_2$,$\cdots$ , $f_n$. Suppose viscous forces are represented by $f_2$. Back to the original question, the Reynolds number (Re) associated with the fluid flow would be, in this case, $m a / f_2$. So, in fluid flow, Re is a measure of the ratio between the resultant force (or inertia force) and viscous force acting on the fluid. Notice that the viscous force is part of the inertia force. In other words, Re is the ratio between the resultant force acting on the fluid and one of its components.
Our intuition regarding effects of inertia forces is quite good because in daily life our muscles overcame inertia of static bodies all the time. Inertia of moving bodies are also easily perceptible when our velocity (for instance in a moving car) changes magnitude or direction, that is, when we are accelerated relative to the ground (notice that a is a factor in the inertia force). Effects of viscous forces are much more subtle and require specific experiments.