"The chase" is first mentioned when Einhard recounts how Charlemagne's children were educated.
the plan that he adopted for his children's education was, first of all, to have both boys and girls instructed in the liberal arts, to which he also turned his own attention. As soon as their years admitted, in accordance with the custom of the Franks, the boys had to learn horsemanship, and to practise war and the chase, and the girls to familiarize themselves with cloth-making, and to handle distaff and spindle, that they might not grow indolent through idleness, and he fostered in them every virtuous sentiment.
It is mentioned again later:
In accordance with the national custom, he took frequent exercise on horseback and in the chase, accomplishments in which scarcely any people in the world can equal the Franks.
Do we know what the chase was? Was it simply pursuing a target (deer, man, etc.)?
It can mean any kind of hunting. But for the period, for noble born men and women, it would typically be with dogs and horses following the dogs, somewhat like fox hunting is practiced in modern times.
Hunting in the middle ages was considered "educational" because it teaches riders to handle their horses in rough terrain and under a certain degree of stress. This is obviously quite a useful military skill for the time. (Hawking much the same, but of course it requires hawks rather than hunting hounds.) This is partially why it was so popular among the French and later English noblity.