Im Federmäppchen liegt ein blauer Füller neben dem Radiergummi.

Im is a contraction of in dem.

• in = in
• dem = the (dative form)

So Im Federmäppchen means in the pencil case.

German very often contracts:

• in dem → im
• an dem → am
• zu dem → zum

So Im Federmäppchen is completely normal and natural.

Both words exist, but Federmäppchen is very common for pencil case.

It is a compound noun:

• Feder = feather / pen
• Mäppchen = little pouch or little case

The ending -chen makes a noun diminutive, so Mäppchen is a small case/pouch. Like all nouns ending in -chen, it is neuter, which is why we get dem Federmäppchen.

German often uses position verbs where English just uses is.

• liegen = to lie, to be lying
• stehen = to stand
• sitzen = to sit

A pen is thought of as lying somewhere, so German says:

• Der Füller liegt im Federmäppchen. = The pen is in the pencil case.

Using ist would not be the most natural choice here. liegt sounds more idiomatic because it describes the pen’s position.

The verb agrees with the subject, not with every noun in the sentence.

The subject here is ein blauer Füller.

So:

• der Füller liegt = the pen lies / is lying

The other nouns are just part of prepositional phrases:

• im Federmäppchen
• neben dem Radiergummi

They are not the subject, so they do not control the verb.

Because German is a verb-second language in main clauses.

That means the finite verb usually comes in the second position, but the first position can be filled by different elements.

Here, the sentence begins with the location:

• Im Federmäppchen = first position
• liegt = second position
• ein blauer Füller = subject after the verb

This is completely normal German word order.

You could also say:

• Ein blauer Füller liegt im Federmäppchen neben dem Radiergummi.

That version starts with the subject instead.

Because German adjectives usually take an ending when they come before a noun.

Here:

• Füller is masculine
• it is in the nominative
• it comes after ein

So the correct form is:

• ein blauer Füller

This is part of German adjective declension.

Compare:

• ein blauer Füller (masculine nominative)
• einen blauen Füller (masculine accusative)
• eine blaue Tasche (feminine nominative)
• ein blaues Buch (neuter nominative)

Because neben is a two-way preposition. It can take either:

• dative for location
• accusative for movement toward something

In this sentence, the pen is already located next to the eraser, so German uses the dative:

• neben dem Radiergummi = next to the eraser

If there were movement, you would use accusative:

• Ich lege den Füller neben den Radiergummi. = I put the pen next to the eraser.

For the same reason: in is also a two-way preposition.

• dative = location
• accusative = direction/movement into

Here the pen is located inside the pencil case, so we use dative:

• in dem Federmäppchen → im Federmäppchen

Compare:

• Der Füller liegt im Federmäppchen.
  = The pen is in the pencil case.
• Ich lege den Füller in das Federmäppchen.
  = I put the pen into the pencil case.

Usually Füller means fountain pen, not just any pen.

It is short for Füllfederhalter, but in everyday German people commonly say Füller.

Other common pen words:

• der Kugelschreiber = ballpoint pen
• der Kuli = informal short form of Kugelschreiber
• der Stift = pen/pencil, a more general word in some contexts

So ein blauer Füller is specifically a blue fountain pen.

In German, all nouns are capitalized, not just proper names.

So in this sentence:

• Federmäppchen
• Füller
• Radiergummi

are all capitalized because they are nouns.

This is one of the most noticeable differences from English spelling.

Yes. In dem Federmäppchen is grammatically correct.

But in normal German, im is much more natural because it is the standard contraction of in dem.

So:

• Im Federmäppchen liegt ein blauer Füller. = most natural
• In dem Federmäppchen liegt ein blauer Füller. = correct, but less usual unless you want emphasis

The important one here is ü in Füller.

A rough guide:

• ü is not the same as English oo or you
• it is a front rounded vowel, a sound English does not really have

A common trick:

1. Say ee as in see
2. While keeping your tongue there, round your lips as if saying oo

That gets you close to ü.

Very rough pronunciation help:

• im ≈ im
• Federmäppchen ≈ FAY-der-mep-khen (very approximate)
• liegt ≈ leegt
• ein ≈ ine
• blauer ≈ BLOW-er
• Füller ≈ FYOO-ler (only approximate; the real ü is different)
• neben ≈ NAY-ben
• dem ≈ daym
• Radiergummi ≈ rah-DEER-goom-ee

If you want to sound natural, the biggest pronunciation target in this sentence is probably ü in Füller.