There are a few ways of classifying the constants of nature. Many classifications divide along "intrinsic" vs "extrinsic" quantities, but this isn't too intersting for us. Instead we want a classification that is attuned to fine-tuning arguments. For those not in the know, fine-tuning is when the constants of nature fundamental to a theory seem arbitrary. Fine-tuning problems in physics have often resulted in unscientific anthropic arugments: divining sample spaces over possible universes with various constants and trying to show that the universes that allow for life must have constants near our measured values. It's our hope that by classifying constants according to their relationship to physical theories we can clear some of the mess around fine-tuning problems.
Examples: $c$ (speed of light), $k_B$ (Boltzmann's constant), $G$ (Gravitational constant), T_0 (absolute zero)
Determinate constants are constants that set the scale of a theory - how a theory translates between its own outputs and our measured outputs. For example, in special relativity we would never ask "why is the speed of light so fast?", but rather "why are most things so slow?". At our current state in physics different theories have different determinate constants. These constants are usually set to 1 in the natural units of the theory. Namely, we have that the relationship between elementary charge $e$, planck's constant $\hbar$ and the speed of light $c$ is given by the fine structure constant $\alpha$ with value approximately $1/137$. In the current state of the art we have to deal with the fact that not all of these quantities can be set to 1 - setting any two to 1 gives a value for the third constant. it's possible in a more fundamental theory of physics we will be able to calculate $\alpha$ but such a thing is far from guaranteed.
Examples: $m_p$ (mass of fundamental particles), $\Lambda$ (comsological constant), $\alpha$ (fine structure constant)
Fundamental constants are similar to determinate constants, but they are not set 1 in the underlying theory. Fundamental constants are empirical inputs to a theory. They must be measured and put into relation with those determinate constants. The choices of which constants are determinate and which ones are fundamental is often a matter of theoretical and experimental convenience. For example, in classical electromagnetism, the electric constant $\epsilon_0$ is fundamental, but in relativistic and quantum electromagnetism it is instead a measured quantity put into relation with planck's constant and speed of light.
Fundamental constants are also the usual target of fine-tuning arguments. There is a drive in physics to reduce fundamental constants as much as possible, so that theories have the minimal amount of empirical input they need. Whether a fundamental theory of physics will contain no fundamental constants is yet to be seen. It has been a promise among some string theorists, but the project of predicting fundamental constants seems mostly dead and held alive by arbitrary anthropic arguments.
Examples: $b$ (Wien's constant), $K$ (Coloumb's constant), $R_H$ (Rydberg's constant), $a_e$ (anomoulous electron dipole moment)
Formal constants are determined by the formal apparatuses of a theory. They are generally at first determined phenomenologically and then theoretically predicted later on. They often stand as guideposts for theoretical development and as testatements to the accuracy and versatility of a theory. For example, the prediction of the rydberg constant via quantum mechanics stood as a marker for the utility of the theory in reproducing the structure of the atom in general.
Examples: $d(♂)$ (distance to mars), $R_{mw}$ (radius of the milky way)
Informal constants are often not thought of constants at all, but often used to be thought of constants in the past. They are in principle predictable from a theory, but practically impossible to predict. The fact that there are such informal constants at all should be a sign of the outrageous success of certain theories. E.g. newtonian gravity is so successful that we do not even expect a prediction for these values. These constants are often used as justification for anthropic arguments - namely we cannot predict the distance between the sun and the earth, but since we have a rigorous dynamical theory we can understand how the distance between the sun and earth comes to be and check that it comparts with our experiences. We understand the earth must be a certain distance from the sun to allow for the development of life. Anthropic arguments generally try to move fundamental constants into informal constants by similar sorts of reasoning.
Examples: $\pi$, $e$
Mathematical constants appear everywhere in physics. They are quite hard to confuse with other sorts of constants, but they do often relate fundamental constants to formal constants. For example, Gauss' law tells us how to relate the electric constant to coloumb's constant in a fairly simple way. Mathematical constants can, in fact, speak to physical facts in this manner. For example, the fact that a quantity $4 \pi$ appears in many equations tells us that the geometry of the underyling theory is flat.
Physical theories place their constants in different categories, and breakthrough in physics often result in the movement of constants to different categories. Newtonian physics moved many fundamental constants to informal constants. The entire 20th century of physics saw a slow move towards the creation and understanding of determinate constants. Depending on theoretical simplicity, determinate constants can move to fundamental constants and vice versa. Movement throughout all of these categories, with the exception of mathematical constants, is common throughout the history of physics.
One reason, among many, to distrust anthropic arguments is that they seek to move fundamental constants to informal ones - not as a result of a strong underlying theory, but rather as a justification for a theory, usually string theory. However, informal constants do not bolster the utility of a theory, they are in fact taxing to the utility of a theory - it is only through great progress in understand nature that we can be confident that certain quantities are not predictable. The most useful theories turn fundamental constants into formal and determinate constants. The existence of informal constants is the price we have to pay for using simple ideas to describe a messy universe. Hopefully, future successful theories of physics will do just that, and any informal constants they create will be a byproduct of rigorous understanding rather than an ad-hoc and arbitrary mess.