Unchanging Objects
The mass of an electron is 9.11 x 10^-31 kg, and its charge is -1.6 x 10^-19 coulombs. Those are constants. They're constants because an electron has the same properties all throughout the universe. So if anyone, anywhere, anytime measures the properties of an electron, those will be the measured values. The same thing is true for other physical entities: protons, neutrons, hydrogen atoms and more.
Relationships
Electric and magnetic fields are related to each other. One way of making an electric field is by changing a magnetic field, and changing an electric field creates a magnetic field. Radio, light and X-rays are all examples of that kind of situation: a decreasing electric field causes the magnetic field to increase, and a decreasing magnetic field causes the electric field to increase. That's why an electromagnetic wave can keep going forever. The relationship between those two fields is a constant --- it's the same wherever it's measured. The constant relating the two fields is the speed of light.
The Strength of Forces
Newton determined the form of the force of gravity. Any two given objects exert a gravitational attraction on each other, no matter how far apart they are. The magnitude of the force is proportional to the product of the two masses divided by the square of the distance between the two objects. The form of the equation is
Force = G x (mass1 x mass2) / distance^2. G is a constant of proportionality. Specifically, it's the gravitational constant, and it's equal to 6.7 x 10^-11 m^3 / ( kg sec^2).
Inconstant Constants
You can do an experiment to measure the gravitational constant, and the results would be duplicatable. Newton's equation for gravitational attraction is universally true, unless the given object is extremely heavy or moving extremely fast. The gravitational constant is a constant everywhere --- except very close to stars or for objects traveling at almost the speed of light. This is an example of a constant with exceptions.