(1) The most successful constructed language so far. It's aimed to be an easy-to-learn and culturally neutral language for international communication. It has, however, been criticised for being too euro-centric and incorporating lots of grammatical features (case marking, agreement of adjectives, several participles) that are quirky to speakers of many non-European languages.
(2) A general term for something artifactual or unnatural, without foundations in the culture of ordinary people.
(2) The Swedish voters rejected the esperanto currency (the Euro) in a referendum.
(1) To a linguist, grammar is the structure of a language (as opposed to its lexicon, its words). Contrary to popular opinion, grammar is not something enforced by language academies, but it exists necessarily in all languages.
(2) To a non-linguist, grammar usually refers to various pre-scientific notions of "correct" usage.
Chinese grammar is very different from English.