Let's think about the verb помогать/помочь. It conjugates like this:

The main meaning of the verb is “to help, give aid to,” and the person receiving the help appears in the dative case. The action which the subject helped the indirect object to do appears in an infinitive phrase:

Let's think about the verb ложиться/лечь. The first thing to notice is that it is one of only four verbs in Russian where the imperfective is reflexive and the perfective is not.¹ It conjugates like this:

The primary meaning of the verb is “to lie” in the sense of “to lie down.” The first thing to notice is that the thing you lie down on appears in the accusative case after the preposition на:

It's also possible to lie under something под + accusative:

We often find this verb in is in combination with спать. The resulting phrase means “to go to bed.”

¹ You get extra points if you can name the three other verbs. Lots of people can name two others, but if you get three others, you are special. Add a comment to this post to prove your prowess!

² The verb “to lie” in the sense of “to lie down” is currently shifting in English. The old standard for the past is to say “Yesterday I lay on the couch after lunch because my head hurt” and the standard for the infinitive can be seen in “My head hurts, and I need to lie down.” In modern American English now people often say “Yesterday I laid down on the couch after lunch” and “My head hurts, and I need to lay down.” This verb is one my personal pet peeves in life, and I'm going to defend the old literary norm either to my dying die or to the day my sister finally admits she has been saying it wrong all these years, whichever comes first.

Death seems to be a-comin’ awful fast.

Let's think about the verb готовить/приготовить, which conjugates like this:

The verb pair готовить/приготовить has a couple meanings, the first of which is “to prepare.” The person/thing who is being prepared appears in the accusative case, and the thing for which they are being prepared appears in the dative case after the preposition к:

The second meaning of the verb is “to cook”:

How do Russians deal with the last names and first names of foreigners? The general rule makes sense: the name declines if it ends in a sound that is typical for a male/female name and if it matches the gender typical of such names.

Not too surprisingly, if the foreigner is male, and if his first name and last name end in a consonant typical of a Russian masculine first declension noun, then the first and last names decline just like masculine first declension nouns. For example, Brad Pitt's name declines like this in Russian:

Likewise if a female foreigner's names end in a letter typical of a second declension noun, the names decline like second declension nouns. Thus Christina Aguilera's name declines like this in Russian:

If either of a man's names ends in a sound that is not typical for a masculine first or second declension Russian noun, then that name is not declined. The other name may still decline. Here's how the names of Antonio Banderas and Gérard Depardieu work:

If either of a woman's names ends in a sound not typical for a second declension noun, then that name does not decline. The other name may still decline. As example we see Paulina Rubio and Jane Fonda:

If a foreign name ends in a sound atypical for Russian male and female names, then usually the name doesn't decline at all, e.g. the names Вупи Голдберг, Мелани Гриффит, Пинк, and Антонио Сабато never change their endings.

A few general examples:

(Can you believe that Julio Iglesias, the idol of menopausal women everywhere, even has an official website in Russia? Click here to immerse yourself in the degradation.)

The generalizations we just discussed take care of most foreign names in Russian. There are lots of other possibilities, but they are too many to enumerate in a short blog entry. Suffice it to say that every once in a while names surprise you.

Quirky exceptions:

• Heaven knows why, but—in terms of official written Russian stylistics—Polish last names in -ski are usually transliterated into Russian not as -ский but as -ски, and they do not decline, thus «фильмы Романа Полански» “the films of Roman Polanski.” If you are writing a formal report in Russian or giving a formal presentation, that's how you have to treat them. But those rules don't hold up later in the day. Even overeducated native speakers of Russian with a fetish for film will say фильмы Романа Поланского when socializing after their big presentation.
• Likewise, although first and last names in -а are generally declined, officially speaking last names that end in -иа are not declined in überformal contexts, thus you'll hear «книги Джона Гарсиа» “the books of John Garcia” if someone is making a formal presentation, although afterwards, when their students are fawning over them at the parties after the conference, the same speaker will say книги Джона Гарсии.

One last thought: the Russians themselves are not always sure how to handle foreign last names and whether to decline them or not, or even how to spell them. When I looked at the Russian Wikipedia entry today for author John David Garcia, I saw that in the title of the page they spelled his name Гарсиа (standard transliteration of the name), but in the body of the article they tended to write Гарсия.

Today let's think about the verb отвечать/ответить. Notice that it's imperfective form is a first conjugation verb, and its perfective form is a second conjugation verb:

The first meaning of the verb is “to answer, respond”:

If you answer a person, the person appears in the dative case:

If you answer a question, then вопрос is the object of the preposition на in the accusative case:

If you combine both the person to whom you are responding and the question, then you have two options. The person can appear in the dative case outside of the на phrase, or you can turn the person into a possessor of the question inside the на phrase: