I would say there is not a large difference to the English language. There are so many ways to express human movement because we do not always move in the same way. Therefore, there must be verbs which differentiate in how fast or slow the movement is or how our body posture looks like.
This is the usual way of human movement. It can mean "go", "go out" or even "leave".
Ich gehe nach Hause. – I am going home.
Weißt du, wohin sie geht? – Do you know where she is going?
Bald muss ich gehen. – Soon I have to leave.
Die Jäger gingen zur Jagd. – The hunters went out for a shoot.
This is a faster movement, meaning "run" or "walk fast".
Ich musste heute Morgen laufen, um den Bus noch zu erwischen. – I had to run to catch the bus this morning.
Er lief so schnell, dass ich kaum mit ihm Schritt halten konnte. – He walked so fast I found it hard to keep up with him.
In some regions, e. g. Braunschweig and Hannover, "laufen" is used as a synonym for "gehen" in a casual context , especially if you have to go on foot. This is also listed in the Duden (1c). Example:
Weil mein Auto kaputt ist, muss ich morgen zu Fuß zur Arbeit laufen. – Since my car is broken, I have to walk to work tomorrow.
This is similar to "laufen", but it generally emphasizes the high speed even more. You can translate it with "race", "run" or "sprint". However, the verb is used more for everyday running. For sporty running you use "laufen" though.
Sie rannte, so schnell sie konnte. – She sprinted as fast as she could.
Ich rannte den Bahnsteig entlang, um den Zug zu erreichen. – I sprinted along the platform to catch the train.
Er kann sehr schnell rennen. – He can run very fast.
This actually means "hike", "walk" or "wander". Hence, it means "walk without a particular destination" or "go leisurely". It can also symbolize migration.
Ich wandere gern im Harz. – I love hiking through the Harz.
Er wanderte den Kamm des Hügels entlang. – He walked along the crest of the hill.
Sie wanderte am Strand entlang. — She wandered along the beach.
Wenn ich nicht schlafen kann, wandere ich im Haus herum. – When I cannot sleep, I roam the house.
This is what you usually do in a park, for example – "stroll", "stroll around" or "walk".
Der Junge spaziert mit seinem Hund durch den Park. – The boy is strolling through the park with his dog.
Wir spazierten/gingen spazieren, obwohl es regnete. – We went for a walk although it was raining.
Nach dem Frühstück gingen wir am Strand spazieren. – After breakfast we took a walk on the beach.
This is just a rough overview. I will not go into details or refer to special expressions here, because this would definitely go beyond the scope. It certainly makes sense to take a look at the Duden or another dictionary.
 Please note Hubert Schölnast's comment bellow.