The verb "apprise" has evolved from its earlier use in a more specific context of estimating value, especially in terms of property or goods. In today's language, it is primarily used in contexts where communication, particularly in terms of important information, is the focus. When you "apprise" someone of something, you are giving them an update or alert, which could range from a simple notification to a detailed briefing.
While it shares similarities with synonyms like "inform" or "notify," "apprise" often carries a slightly more formal or deliberate tone, particularly in legal, business, or governmental contexts.