Israel: the name given to Jacob, and then to the covenant people and nation descended from him
Israel is first the new name given to one man, Jacob, after a night of wrestling at the Jabbok river, and then the name of everyone descended from him. The whole life of the word is a steady widening of who it points to. It starts as a personal name for a single patriarch, becomes the name of his family grown into a people, then the name of a kingdom, then (after that kingdom split in two) the political name of just the northern half over against Judah in the south, and finally a deep word for the whole covenant community of God, an identity that outlived both the land and the monarchy.
These are not different words but one name aimed at a moving target. The shared thread is always the covenant line that runs from Jacob outward: the senses below simply mark how far down that line a given verse is looking. In Genesis the name usually means the man himself. From Exodus on it almost always means the people, often in the set phrase the children of Israel. In Kings and the eighth-century prophets it frequently narrows to the breakaway northern kingdom. In Isaiah of the exile and in Ezekiel it widens again into the chosen servant people and the reunited nation of the future.
The famous explanation the Bible itself gives, that the name means one who strives with God, comes from the wrestling story and is a deliberate play on the sound of the name rather than a strict translation of it. Most scholars think the name is older than that story and was originally a sentence about God acting, something like God rules or God contends. The name therefore holds two truths at once that the Hebrew Bible never fully resolves: God acts, and this people struggles. The etymology debate is laid out below.