A chemical equilibrium concerns chemical reactions. There should be at least a forward- and backward reaction between two species but more complex systems with multiple individual reactions may occur. The important observation is that there is no macroscopic change to the chemical constituents of the system, i.e. the concentrations of all reaction partners remain the same despite all the individual reactions happening. Chemical equilibria can always be written with reaction equations such as below
$$\ce{H2SO4 + 2 H2O <=> HSO4- + H3O+ + H2O <=> SO4^2- + 2 H3O+}$$
Before I get to dynamic equilibria, it is important that I briefly note equilibria in general. In equilibria, there must be some kind of balance but there need not be any movement. For example, in a tug-of-war, two groups of people are pulling the same rope in different directions. When there is no macroscopic movement, it means that both sides are pulling with the same force. This is an equilibrium. However, even if you go down to a microscopic level, you will still notice no movement (i.e. the rope would seem (mostly) stationary). Thus, this type of equilibrium is classified as a static equilibrium: there is no change to the overall state because there is no change to any individual states.
In a dynamic equilibrium, the macroscopic state remains the same but if you care to look at the individual microscopic parts you will notice them switching between states. For example, consider two sealed rooms, one filled with nitrogen and one with oxygen separated by a door. Once that door is opened, gases can flow back and forth. After a certain time, the concentrations in both rooms of both gases will have equilibrated (say to a $4:1$ ratio if that is the ratio of room sizes). Individual gas molecules will still flow back and forth through the open door, but as their concentrations are identical on both sides the flow rate for individual gases is the same and the constant back and forth will not affect the overall ratio of the rooms. This is a dynamic equilibrium because the individual molecules move back and forth (microscopic change) while the overall state remains the same. However, it cannot be written as a chemical equation and thus it is not a chemical equilibrium.