How does one determine whether one is in an inertial frame?
An inertial frame is one on which a particle with no force on it travels in a straight line.
But how does one determine that no forces are acting on it? After all, one might declare a force is acting on a particle because it is deviating from a straight line.
Imagine a point mass attached to ends of six identical springs of relaxed length $L$. The springs are oriented in co-linear pairs, with these pairs mutually perpendicular to the other pairs, i.e., they form an $(x,y,z)$ coordinate system. Attach the other ends to the walls of a cube of dimension $2L$.
If the mass remains in the center of the cube, the cube is in an inertial reference frame.
There are a certain number of known forces, so you could start by ruling each one out by experiment.
Otherwise the so-called fictitious forces - the centrifugal force and the Coriolis effect - have the characteristic that they provide the same acceleration to all objects, regardless of their mass. As you know, in classical mechanics $a = F/m$ so forces will in general accelerate objects with different masses at different rates. The only known real force that does that is gravity.
Bonus material: This last observation was in fact central in developing General Relativity. In GR, gravity is indeed seen as a fictitious as the centrifugal force, and due to the way the observer lays the spacetime coordinated.
An inertial frame of reference is one that is not accelerating. A frame that is accelerating with respect to another inertial frame is a non-inertial frame of reference.
An inertial frame moves with steady velocity in respect to other inertial frames, such that an observer within the frame can not detect any movement of the frame, unless she looks outside the frame. There is no difference in the attributes of any direction within the frame.
A clock placed in any part of the inertial frame can be synchronized with a clock in any other part of the frame, and all clocks so synchronized tell the same time.
If a particle deviates from a straight line, start looking for forces within the inertial frame that would cause the deviation. If you can't find any, you quite possibly are NOT in an inertial frame of reference. However, even though an object deviates from a straight line in a gravitational field, you may consider the surface area of the gravitating body to be an inertial frame of reference, as the normal force between the surface and objects on the surface cancels out acceleration.
Short answer:
One cannot determine whether a reference frame is inertial. It is simply not possible. This situation is avoided in General Relativity (or any of its extensions), since in GR the physics does not depend on the reference frame (any reference frame), and the concept of inertial frame is not necessary. Of course one can determine an inertial frame which is good enough for a given system, but one cannot determine, for example, an inertial frame to describe the motion of galaxies in the whole universe.
Long answer:
As pointed out by Andrea Di Biagio, one can reason by exclusion. If a particle moves on a straight line and no force is applied, the reference frame is inertial. Since there is a finite number of known forces, one can in principle rule out each of them and reduce to a situation where no force is applied to the particle. There are at least two weak points in this reasoning:
1) The list of known forces in modern physics is: gravitational, electromagnetic, weak and strong interactions. No one knows if this list is complete of course, therefore in principle reasoning by exclusion cannot work.
2) In classical mechanics, two distant bodies can interact over a long distance through the electromagnetic or gravitational field. Therefore to exclude, say, electromagnetic and gravitational forces, one should in principle know the charge and mass distribution of the entire universe. Consider the following situation. In a hypothetical laboratory in deep space, far away from galaxies and other visible mass densities, a neutron is observed to travel in a certain reference frame. Is this reference inertial? What if a very massive structure is placed just beyond the portion of universe observable by the laboratory? Will this mass distribution exert a force on the neutron in the laboratory? As one can see, the concept of inertial frame is very problematic, and the paradoxes which arise from this concept are not curable in the framework of classical (Newton) mechanics.
I asked someone who studied physics this question and was told that the following is a definition:
A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown is called an inertial frame.
