Grammar (how are words constructed and orderly connected? what classifications / parts of speech does English / Esperanto have?):
It is extremely like Japanese, where there is a root word (that has no stem-change) which simply changes in ending or prefix according to which role in grammar it is playing, and where verbs can (and often do) compound onto each other. Also it's the same in where there are very very few exceptions. This basic concept is modernly very different in English, as for example, we have many mismatching pairs like "city, urban", "rain, pluvial", "fear, coward/poltroon, scary" and so on. In Esperanto every single pair stems from the same root, and there are a few more suffixes than in English, ex. one for "to cause" and one for "to become" (this was present in Old English and we have remnants of it today, ex. in bedazzled). For example:
- There is no "famous, red, blue, wet, dry, lethargic, celestial", as in Esperanto they're "fama, ruĝa, blua, malseka, seka, letargia, ĉiela". All are adjectives in both languages.
- "angrily, wetly, very, lethargically, on Mondays" are "kolere, malseke, tre, letargie, lunde". All are adverbs.
- "sincerity, hound (dog), dilapidation, epoch, eczema, shoulder-blade" are "sincereco, hundo, ruinigo, epoko, ekzemo, skapolo". All are nouns.
- "pony, puppy, kitten, duckling" are "ĉevalido, hundido, katido, anasido" (ido means "an offspring", so it's like "baby horse").
And so on and so forth. Changing the prefix or suffix is the way to get the corresponding related pair (unless you want to use a synonym, but this is the normal way). Kolere is angrily, kolero is anger. Ĉiela is celestial, ĉielo is sky. That sort of logic is the basis of the entire language. Patro is father, bopatro is father-in-law.
Words are basically constructed in the same way as English - the starting word of a compound modifies the ending word of the compound (an oft-cited example is "songbird and birdsong"). Most of the prefixes and suffixes have some sort of equivalent in English too. The difference is that in Esperanto, practically any word can be compounded — this includes prepositions being compounded to each other, or prepositions being prefixed to verbs etc. (as is done in Swedish, among other languages).
For example, "to pick out" can be said in exactly that word order, but "to outpick" can also be said. Prefixes and suffixes can also be compounded to each other, for an easy example pra-avo (grand-pa), pra-pra-avo (great-grand-pa). Numbers can also be compounded into words, ex. unu-op-ulo ("one-collective-being/person", meaning "individual").
As for word order in general, it has very little rules. The definite article and the word "no" have to come before the word they describe (but the adjective that describes the noun, can come either before or after the noun, for example). Words marking questions (like "does" in "Doesn't he want to eat?") have to be the very first word in the sentence or clause that introduces the question. The rest of the so-called word-order rules are basically just trends, people have decided that one trend or another gives a certain flavour and so on. The same is true for compounds, in that it's just usually a trend as to whether a word is written as a compound or written separately, both are allowed in the language.
So in Esperanto it's perfectly acceptable to have sentences just as in English, with "I eat apples on Monday", but also perfectly acceptable is as in Swedish, "On Monday eat I apples", or even bigger variations like "I apples on monday eat". This is thanks to an accusative case existing, but even then, the rule is that "the accusative doesn't have to be used if the meaning is obvious".
As for verb tenses and moods, everything that can be said in English is perfectly able to be said in Esperanto. The difference is that Esperanto speakers tend to not use all of the more precise forms. For example, Esperanto can easily say "I had been going", but instead people will simply say "I went". This mirrors Japanese, Swedish, etc. In a similar vein, Esperanto verbs do not conjugate according to number or gender, so it essentially looks like "I am, he am, they am".
The parts of speech are just the same as in English if you ignore that it has the accusative case, and there are a few items that are simply more clarified than English. There are nouns, adjectives, adverbs, exclamations, prepositions, conjunctions and "particles" (I forget what they're called - words that aren't really any of these categories). There's present, future, and past tense, also the subjunctive and command form; other forms like "I want to go" are expressed with compounds or prepositions etc. The more clarified parts are that Esperanto has a distinct word for reflexive pronouns, like "I did it to myself", versus "I don't like it, myself". The only cases are nominative and accusative, but there is effectively a genitive because the same ending for adjectives is also the possessive ending (though we can also denote a posession using "of", as in English "the hair of the dog"). There is no indefinite artcle in Esperanto (which mirrors Icelandic, Japanese, etc) but if necessary we can say "one dog", for example.
Vocabulary (from what languages does English / Esperanto borrow terms / derived words?):
Originally, most words were from Latin and German. This branches into similarities with languages such as French, Yiddish, English, and Danish.
Modernly, Esperanto borrows either the most international word, or the only word that can be borrowed (ex. "geisha" is of course only available from Japanese). In some cases, authors took words from languages they knew and so even if there were more international words, those words became the common ones (some vulgar words are like that). So, just as with where most of English's loanwords come from, most of Esperanto's new/loanwords nowadays come from Latin or Greek. Most technology words are therefore basically the same in both languages, such as "television" or "internet", where there's just one or two letters in difference between the languages. The Esperanto is "televidilo" (a compound of "tele- + see + tool"), as the Latin roots themselves are tele- and videre (to see); for internet it's "interreto". Medical and botanical/animal words however are pretty much always borrowed from Latin. There are a few trends like that how most time words (day, month, year, minute - tago, monato, jaro) were all originally borrowed from German.
When Esperanto doesn't borrow words, it simply makes them up from existing roots. For example, bildliteraturo means comics, and bild is picture in German, Swedish, etc. Of course, literature comes from French and Latin (in Latin it's literatura, apparently). The same word in Swedish would be "drawn series", and in Japanese for example the compound includes "picture/painting" instead of "literature", so as you can see Esperanto creates the word from its most logical form instead of directly borrowing the way another language created it.
Semantics (words in what languages share similar connotations and implications with English / Esperanto words?)
There are a great many of these in base/root words, thanks to English's large vocabulary and the fact that, of course, no language uses words completely literally and everything can be a metaphor or implication anytime one wishes. For example:
pluvial - pluvia
hound (dog) - hundo
paper - papero
to comprehend - kompreni
The main difference is that, once a word becomes a compound word, it usually ceases to look the same as English. Also there are cultural implications of certain words that are different from country to country, so they aren't always the same in Esperanto. This is especially true because Esperanto is least known about in the English-speaking world, and English's effect on Esperanto is actually very minimal compared to most other languages (ex. only 20% of translations into Esperanto come from English; other languages tend to be more like 60-80%). But here are synonyms with shades of meaning in Esperanto that correspond to synonyms with shades of meaning in English, yes.
You didn't ask about it, but I figure you're probably wondering - Esperanto has made "neologisms", slang, and sometimes has its own special implications on words. For example, a "green pope" is someone who talks about Esperanto constantly (green is the "colour of Esperanto"). A "necessary-place" is a toilet/bathroom. There are also verbs and so on named after people, this is equivalent to saying "he's a Marxist", or "a McCarthy-ism", where the name of a certain famous person has taken on a particular meaning; only of course Esperanto has its own famous Esperanto speakers it takes the names from. Outright slang words are relatively few as it would complicate the language too much, but there are also unique meanings strongly implied with certain words that are only known to people who know Esperanto and/or its history (which, frankly, is most Esperanto speakers).
As mentioned by the other person who replied, if you really want to know it you have to learn it (you don't have to become fluent even). It's honestly very fast to learn, all you need to do is get a dictionary like this one which has a list of prefixes and suffixes, then look up words on the online dictionary on lernu.net (which often breaks them into their compounds too).