Degree of success is a measure of a character's success during resolution of a conflict or moment of uncertainty. It is connected to the subsequent effect of the character's actions as a result of their success (or failure or otherwise), but the term degree of success generally refers to a mechanic's distinct success-states at the execution of an action, rather than the subsequent impact on the game world that is determined by a resolution's effect.
Types of degree of success
The most basic collection of success-states are a binary success or failure. With success, the character accomplishes what they intended; with failure, they do not.
In addition to binary success/failure, many systems include the possibility of critical success (often called critical hit when referring to attacks) and/or critical failure (aka fumbles). A critical success is a success that is particularly effective, usually by improving the subsequent effect (e.g. increasing the damage done by an attack in games like Dungeons & Dragons). A critical failure is a failure that not only does not accomplish the character's goals, but has a negative impact for the character making the attempt.
In the language of improv theatre, a critical success can be considered "Yes, and" whereas a critical failure can be considered "No, and".
Partial success and success with cost
Some games, e.g. Fate Core or those Powered by the Apocalypse, have a middle possibility between outright success and outright failure, which is called a partial success, mixed success, or success with cost. This is generally a success, but one that comes with additional costs for either the character or the player. Sometimes the player will have the option of declining the offer and failing if they are not willing to pay the cost, but in other games the outcome is taken as read.
In the language of improv, a partial success can be worded "Yes, but".
Some games have different levels, tiers, or degrees of success. For instance, Torg: Eternity has a success, a Good Success, and an Oustanding success. These degrees may be defined as certain thresholds, or a game could have a theoretically unlimited number of degrees of success based on how much the difficulty was overcome by the action.
Opportunities and threats
Games such as the Star Wars Roleplaying Game by Fantasy Flight Games and the Genesys system that developed from it, use a type of resolution in which opportunities and threats are independent of success and failure. Using custom dice, players can roll and tally up successes, failures, opportunities and threats. Success or failure is determined by which of the two appears most, but there can also be an opportunity or threat (again based on which of the two appears most on the dice) regardless of which outcome the player got on the overall action. Importantly, because opportunities and threats are independent of success and failure, they can't interfere or adjust the effect of a success or failure (i.e. a success with an opportunity shouldn't improve the effect of the success, unlike a critical success in other games).
In the language of improv, this allows for all four possibilities: "Yes, and", "No, and", "Yes, but", "No, but".