How do I discipline a child without yelling or punishing? Children often don't understand what we try to tell them...

Consequences . They will learn they need to listen and follow your rules if not doing so caused for them to lose something they truly like.
When you're both . calm, tell him AT HIS LANGUAGE LEVEL the behavior you expect, the immediate reward for compliance & the immediate consequence for non-compliance. Make sure he understood. Follow through consistently with the walk-don't -talk time-out technique rom "Little People" by Christopherson. Ignore a behavior to extinguish it. First it escalates to try to elicit negative or positive attention, then stops.
Seek to understand. The child first. Approach situations from the child's perspective and gradually help the child learn what is okay and what is not okay using that specific language. Discipline is teaching. Make it fun. Model behavior for the child to accompany the words you use. A positive parent-child relationship is the foundation for child development and the child's self-esteem is vital. Best regards.