Learn Russian: На столе тарелки и чашки.

Both тарелка (plate) and чашка (cup) are feminine nouns ending in -а. Normally, their plural ending would be -ы. However, Russian has a strict spelling rule: after the letters к, г, х, ш, щ, ж, and ч, you must always write и instead of ы. Since both words have a к before the ending, they become тарелки and чашки.

There is no word for 'are' in this sentence. Why is that?

In Russian, the present tense of the verb 'to be' (am, is, are) is almost always skipped. So instead of saying 'On the table are plates and cups,' you literally just say 'On table plates and cups.'

Why is there an -е at the end of столе?

Стол (table) is in the Prepositional case here because it follows the preposition на (on) to show where something is located. For most masculine nouns ending in a consonant, you add an -е to make this form.

Could I say Тарелки и чашки на столе instead?