بقية الطعام في وعاء الطعام.

A careful reading is:

baqiyyatu ṭ-ṭaʿāmi fī wiʿāʾi ṭ-ṭaʿāmi

A few pronunciation notes:

• بقية is baqiyya(tu). Because it is connected to another noun in an iḍāfa phrase, the ة is pronounced as t here.
• الطعام is pronounced aṭ-ṭaʿām, not al-taʿām, because ط is a sun letter, so the l of ال assimilates.
• وعاء is wiʿāʾ. The final ء is a hamza, so there is a little catch at the end.

With full case endings, the sentence is:

بَقِيَّةُ الطَّعَامِ فِي وِعَاءِ الطَّعَامِ

Because Arabic often uses a nominal sentence with no present-tense form of to be.

So:

• بقية الطعام = the subject
• في وعاء الطعام = the predicate, meaning in the food container

So the sentence literally looks like:

The rest of the food — in the food container

But in natural English, we say:

The rest of the food is in the food container.

Literally, بقية الطعام means:

• بقية = remainder, rest
• الطعام = the food

Together: the rest of the food

This is an iḍāfa construction, often called a genitive construct:

• first noun: بقية = the thing possessed or defined
• second noun: الطعام = what it is connected to

So Arabic expresses the rest of the food as rest-of the-food.

Because in an iḍāfa construction, the first noun normally does not take ال.

So you say:

• بقية الطعام
• not البقية الطعام

Important rule:

• the first noun in iḍāfa does not take ال
• the whole phrase can still be definite if the second noun is definite

Since الطعام is definite, بقية الطعام means the rest of the food, not just a rest of food.

Because Arabic iḍāfa often corresponds to several natural English patterns, including:

• of phrases: container of the food
• noun + noun compounds: food container
• sometimes a more idiomatic translation: food bowl

So وعاء الطعام literally is container of the food, but in English we often prefer food container or food bowl, depending on context.

Because the two phrases do different jobs:

• بقية الطعام = the rest of the food
• وعاء الطعام = the food container

So the first الطعام tells you what the rest belongs to, and the second الطعام tells you what kind of container it is.

Arabic often repeats nouns where English might sometimes avoid repetition. If the context is already clear, Arabic could also shorten it, for example:

• بقية الطعام في الوعاء = The rest of the food is in the container
• بقية الطعام في وعائه = The rest of the food is in its container

With full vowel endings, the sentence is:

بَقِيَّةُ الطَّعَامِ فِي وِعَاءِ الطَّعَامِ

Here is the grammar:

• بقيةُ: nominative, because it is the subject of the nominal sentence
• الطعامِ: genitive, because it is the second term of the iḍāfa
• في: preposition, in
• وعاءِ: genitive, because it comes after في
• الطعامِ: genitive, because it is again the second term of an iḍāfa

In normal Arabic writing, these short vowel endings are usually not written.

For the same reason as بقية: it is the first noun in an iḍāfa construction.

So:

• وعاء الطعام literally = container of the food
• because الطعام is definite, the whole phrase becomes definite

That means وعاء الطعام can mean:

• the food container
• the container of the food

So a noun in iḍāfa can be definite without having ال on the first word.

Yes, very often you can.

Both can mean something like:

• the rest of the food
• the remaining food

But there is a slight difference in feel:

• بقية الطعام focuses on the remainder or rest
• باقي الطعام literally means the remaining food

In many everyday contexts, they are very close in meaning.

Not necessarily. وعاء is a fairly general word meaning:

• container
• vessel
• bowl

The exact English translation depends on context.

For example, if you wanted to be more specific, Arabic might use other words such as:

• صحن = plate/dish
• علبة = box/container
• قدر = pot

So وعاء الطعام is a general, perfectly acceptable phrase, but the best English word may vary with the situation.