Hold on means you want somebody to wait and is a command. Example: Hold on, I want to finish my drink. (That means you want your friend to wait for you to finish your drink and then you will do something.) To wait means to not go or do an action. I will wait until I have more money to travel. We have to wait for a table at the restaurant.

“Hold on” is a phrasal verb that can be informally used to mean “wait.” It is often combined with a period of time, for example, “Hold on a minute, let me do this first and I’ll be right back.” It has the connotation of a temporary pause, as in the above example; the speaker is asking the listener to please wait for a short period of time while the speaker attends to something else, after which the speaker intends to return to the initial situation. In formal settings, “wait” is typically more appropriate. Also, “hold on” when used this way is almost always used in this present imperative sense, perhaps sometimes in the future sense (“I’ll hold on for a second while you do that”), but to me it would be quite unusual to hear it in the past tense, and “waited” would be more appropriate. Of note, “Hold up” is similar to “hold on” and is even more informal. (It should be mentioned that “holdup” as a noun is a term that can refer to a robbery, typically at gunpoint or at the threat of immediate violence, and refers to the phrase “hold up (your hands) typically said by the robbers to their victims. Alternatively, “holdup” as a noun can more peacefully refer to a delay; as in, “I’m going to be late, there’s been a holdup.”) When not being used in the above phrasal meaning, “hold on” literally means to continue holding or clinging to something; I would translate it as “aferrarse”; often in the context of holding on to something in order to avoid falling. For example, “We’re heading for a sharp turn; hold on!” Hope this helps!