A field is something that has a magnitude and a direction at every point in space. Gravity is a good example - we know there is an acceleration due to gravity of about 9.8 m/s2 down at every point in the room. Another way of saying this is that the magnitude of the Earth's gravitational field is 9.8 m/s2 down at all points in this room.
Gravitational field: g = F/m
where F is the force of gravity.
We can draw a field-line pattern to reflect that, near the Earth's surface, the field is uniform. The strength of a field is reflected by the density of field lines - a uniform field has equally-spaced field lines.
If we zoom out and view the Earth from far away, we get a non-uniform pattern. In fact, the pattern is radial - the lines are further apart as they get further from the Earth, reflecting the fact that g decreases with distance. At every point, though, the field-line pattern shows the direction of the gravitational force that would be experienced by a mass placed at that point.
How does g depend on distance?
Isaac Newton is probably most famous for the falling apple story, which supposedly led to his development of the Law of Gravitation. This states that for two masses, m and M, separated by a distance r, the gravitational force exerted by one mass on the other is:
F = -GmM/r2
G is the universal gravitational constant.
G = 6.67 x 10-11 N m2 / kg2
The minus sign in the force equation indicates that the force is attractive, and the unit vector indicates that the force is along the line connecting the two masses.
The masses exert equal and opposite forces on one another.
Since the gravitational field g = F/m, in general g has a magnitude of GM/r2 and points toward M. This gives us the value of 9.8 m/s2 we've been using all along:
M = mass of the Earth = 5.98 x 1024 kg
R = radius of the Earth = 6.37 x 106 m
g = 9.83 m/s2