Kalemin ucu kırıldı.

Because kalemin ucu is a possession structure in Turkish, meaning the pencil's tip or the tip of the pencil.

In this structure:

• kalem = pencil
• kalemin = of the pencil
• the ending -in is the genitive ending

So kalemin shows that pencil is the thing that owns or has the tip.

The base word is uç, which means tip, end, or point.

Here it becomes ucu because it takes the 3rd person possessive suffix, which is used in this kind of construction:

• uç = tip
• ucu = its tip / the tip of it

So:

• kalemin ucu = the pencil's tip

Also, the final ç changes to c before the vowel-initial suffix. This is a common sound change in Turkish:

• uç → ucu
• compare similar patterns like ağaç → ağacı

It is a very common Turkish pattern called the genitive-possessive construction.

The pattern is:

• possessor + genitive
• possessed thing + possessive suffix

So here:

• kalem-in = of the pencil
• uç-u = its tip

Together: kalemin ucu = the tip of the pencil

This pattern is used all the time in Turkish:

• evin kapısı = the door of the house
• arabanın rengi = the color of the car
• kitabın kapağı = the cover of the book

Because the verb here is kırılmak, which means to break, to get broken, or to snap depending on context.

In this sentence:

• kırıl- = break / get broken
• -dı = past tense

So kırıldı means it broke or it got broken.

With kalemin ucu kırıldı, the natural English translation is usually:

• The tip of the pencil broke.

Even though Turkish uses a form that can sometimes feel close to was broken, in many everyday sentences it simply means something broke on its own or ended up broken.

Because kırdı and kırıldı are different verbs in meaning and structure.

• kırdı = he/she/it broke something
• kırıldı = it broke / it got broken

Examples:

• Kalemi kırdı. = He/she broke the pencil.
• Kalemin ucu kırıldı. = The tip of the pencil broke.

So kırdı needs someone doing the action to something else.
But in your sentence, the subject is the tip, and the meaning is that the tip broke.

Yes. The whole phrase kalemin ucu is the subject.

The sentence divides like this:

• Kalemin ucu = the pencil's tip
• kırıldı = broke

So the thing that broke is the tip of the pencil.

Turkish usually does not have a separate word for the.

So kalemin ucu can mean:

• the pencil's tip
• the tip of the pencil

Turkish often leaves definiteness to context, and possession structures like this are commonly understood as definite.

Yes. In natural English, The pencil tip broke is also a good translation.

Turkish kalemin ucu literally means the tip of the pencil, but in English we can express that in different natural ways:

• The tip of the pencil broke.
• The pencil tip broke.

Both are fine.

The normal order here is:

• Kalemin ucu kırıldı.

This is the most neutral and natural order.

Turkish word order is flexible, but changing it usually changes the emphasis. For example:

• Ucu kırıldı. = Its tip broke.
  This works if it is already clear which object you are talking about.

You would normally keep kalemin ucu kırıldı unless context makes a shorter version possible.

The genitive ending changes by vowel harmony and sometimes by final sound.

The genitive suffix has forms like:

• -ın
• -in
• -un
• -ün

Since kalem has the vowel e, the correct form is -in:

• kalem → kalemin

Other examples:

• kitap → kitabın
• gül → gülün
• okul → okulun

So kalemin is the vowel-harmonized form.

The possessed noun here takes the 3rd person singular possessive suffix, which also changes by vowel harmony.

Its common forms are:

• -ı
• -i
• -u
• -ü

Since the word is uç, the correct form is -u, so:

• uç → ucu

Again, the ç softens to c before the vowel.

This same pattern appears in many examples:

• kapı → kapısı = its door / the door of it
• renk → rengi = its color / the color of it
• uç → ucu = its tip / the tip of it

A rough pronunciation guide is:

• kalemin → ka-le-min
• ucu → u-ju
  (the c in Turkish sounds like English j in jam)
• kırıldı → roughly kuh-ruh-luh-duh, but with Turkish ı

The Turkish ı is important here. It is not the English i sound. It is a deeper, more central vowel, as in:

• kırıldı = kı-rıl-dı

So the full sentence is approximately:

• ka-le-min u-ju kı-rıl-dı

Yes, very common.

Turkish often expresses this idea by naming the whole object and then the part that belongs to it:

• Arabanın kapısı açıldı. = The car door opened.
• Masanın ayağı kırıldı. = The leg of the table broke.
• Kalemin ucu kırıldı. = The tip of the pencil broke.

So this sentence is a very natural and useful model for talking about parts of things.