Learn Finnish: Lapsi syö jäätelöä. - The child eats ice cream.

In Finnish, when the object of an action is a mass noun (an uncountable substance like water, coffee, or ice cream) and you are consuming an indefinite amount (meaning "some" ice cream), it must be in the partitive case. Adding -ä to jäätelö changes it to jäätelöä.

How do we know whether to add -a or -ä for the partitive ending?

This is due to vowel harmony. Because the base word jäätelö contains front vowels (ä and ö), the ending must also be a front vowel. Therefore, we use -ä instead of -a.

How would I say "The child eats an ice cream" (meaning one whole ice cream cone)?

If you mean a specific, single portion of ice cream rather than an indefinite mass, you wouldn't use the partitive case. You would use the form for a total object, which usually ends in -n: Lapsi syö jäätelön.

Why isn't the last vowel doubled in the verb syö?

The base form of the verb is syödä (to eat). To get the he/she/it form, you take the stem (syö-) and usually double the last vowel. However, because syö- already ends in two different vowels (yö), Finnish rules say we don't add a third vowel. So, it stays as syö.