Learn Russian: У жены есть новая машина.

In Russian, to say someone 'has' something, we use the preposition у followed by the owner in the Genitive case. The dictionary form is жена (wife). Since it ends in -а, its Genitive case ending changes to -ы, making it у жены (at the wife).

Where is the verb for 'has' in this sentence?

Russian does not typically use a verb like 'to have'. Instead, it uses a construction that literally translates to 'at the owner there is the object'. Here, у жены means 'at the wife', and есть means 'is' or 'exists'.

Why are the words новая машина (new car) in their normal dictionary forms instead of the Accusative case?

In English, the car is the object receiving the action of 'having'. But in the Russian 'to have' construction, the thing that is owned is actually the subject of the sentence! Because it is the subject (the thing that 'exists' at the owner's location), новая машина stays in the Nominative case, which is the basic dictionary form.

Can I drop the word есть in this sentence?

No, not when you are stating the simple fact of ownership. You must include есть to state that the wife possesses a car. You usually only drop есть when focusing on a descriptive feature of an object we already know exists, like saying У жены красивая машина (The wife's car is beautiful).