הצלחת שלי ליד הצלחת שלך, אבל המזלג שלי לא שם.

In present-tense Hebrew, the verb to be is usually omitted.

So:

הצלחת שלי ליד הצלחת שלך
literally looks like my plate next to your plate,
but it naturally means My plate is next to your plate.

Hebrew does use forms of to be in the past and future, but not usually in simple present-tense sentences like this one.

The ה־ is the definite article, like the in English.

הצלחת שלי means my plate in the sense of the plate that is mine.
Without ה־, צלחת שלי often sounds more like a plate of mine.

Compare:

• הספר שלי = my book
• ספר שלי = a book of mine

So the article helps make the noun definite.

Because modern Hebrew usually expresses possession with של plus a pronoun:

• שלי = of me / mine
• שלך = of you / yours
• שלו = of him / his

So הצלחת שלי is literally something like the plate of me.

This is different from English, where possessives usually come before the noun: my plate, your plate.

It agrees with the owner, not with the thing being owned.

That is important here because צלחת is feminine, but שלך still depends on whether you are male or female:

• שלך pronounced shelkha = your when speaking to a man
• שלך pronounced shelakh = your when speaking to a woman

The spelling is the same without vowels.

Also, שלי does not change for gender, because I / me has no masculine/feminine distinction in Hebrew.

ליד means next to, beside, or by.

The pattern is:

X ליד Y = X is next to Y

So:

הצלחת שלי ליד הצלחת שלך
= my plate is next to your plate

Another example:

הכיסא ליד השולחן
= the chair is next to the table

Because Hebrew usually leaves out vowel marks, so different words can be spelled the same way.

הצלחת can be read as:

• ha-tsalakhat = the plate
• hitzlakhta = you succeeded (masculine singular, past)

In this sentence, the context makes it clear that it is the noun the plate, especially because it is followed by שלי or שלך, which fit naturally after a noun.

So הצלחת שלי is clearly my plate, not you succeeded my... which would make no sense.

Because לא is negating the statement is there.

המזלג שלי לא שם means:
my fork is not there

Hebrew often uses לא to negate this kind of sentence.

By contrast, אין usually means there is no / there isn't in an existence sense. For example:

• אין שם מזלג = There is no fork there

So:

• המזלג שלי לא שם = a specific fork is not there
• אין שם מזלג = there is no fork there at all

Here שם means there.

So:

לא שם = not there

A useful thing to know is that שם can also mean name, but then it is pronounced differently:

• sham = there
• shem = name

Same spelling, different pronunciation, and context tells you which one is meant.

Yes.

• צלחת is feminine
• מזלג is masculine

In this sentence, you do not see much effect from that, because שלי does not change and there are no adjectives.

But gender matters in other cases. For example:

• הצלחת הגדולה שלי = my big plate
• המזלג הקטן שלי = my small fork

The adjective changes to match the noun’s gender.

אבל means but.

It connects the two parts of the sentence and shows contrast:

• first part: the plate is next to your plate
• second part: the fork is not there

So it works just like but in English.

The comma before אבל is also normal in written Hebrew when joining two clauses like this.

A natural pronunciation would be:

ha-tsa-LA-khat she-LI le-YAD ha-tsa-LA-khat shel-KHA / shel-AKH, a-VAL ha-maz-LEG she-LI lo SHAM

Notes:

• שלך is shel-KHA if speaking to a man
• שלך is shel-AKH if speaking to a woman
• kh represents the Hebrew sound ח / כ, like the ch in Scottish loch or German Bach

So the full sentence sounds roughly like:

ha-tsa-LA-khat she-LI le-YAD ha-tsa-LA-khat shel-KHA, a-VAL ha-maz-LEG she-LI lo SHAM