Game characters are usually animated using a technique called skeletal animation:
(Image source: Valve Software)
Each 3d model has an invisible bone structure (the red and teal lines in the image above). Each polygon of the model is connected to a bone. When you define a motion sequence, you define it as a sequence of rotations of the bones around their connections to other bones. When a bone moves, the polygons connected to it move with it. This allows to define motion sequences without having to change the position of every single polygon. When multiple characters share the same bone structure, they can also share the same animations, so you don't have to recreate each animation for each character (although you might want to make some animations different for characters of different genders, personality types and levels of physical fitness).
Most 3d modeling software supports skeletal animation and have export-formats which are readable by widely used 3d engines. Please do not ask for product recommendations here, because they are off-topic.
However, letting an artist create animations from scratch doesn't always lead to the best results. It's hard to recreate all the subtle nuances of human body-language from memory, so the results often look robotic and unnatural. That's why larger game studios which have the resources for it use a process called Motion Capturing.
Human actors are hired to enact the actions of the game characters. The actors wear special suits during the performance which have highly visible marks attached to them. When they are recorded by a video camera during the enactment, the movement of these marks can be mapped to the bones of the character 3d models. That way the movements of the actors can then be applied to any 3d model which uses the same bone structure. This method usually leads to much more naturally-looking results and when there is a large number of animations to create, it can also be a huge time-saver.
(Image source: Toptear Games)
However, the cost for the equipment needed for motion capturing will likely far exceed your budget, so you will have to resort to defining skeletal animations manually in a 3d modeling program.