Мы обсуждаем наш выпускной день.

Обсуждаем is the 1st person plural, present tense form of the verb обсуждать (to discuss).

• Infinitive: обсуждать (imperfective, to discuss)
• Stem: обсужда-
• Ending for мы (we) in the present: -ем

So:

• мы обсуждаем = we discuss / we are discussing

Russian does not have a separate continuous form, so мы обсуждаем can mean both we discuss and we are discussing depending on context.

Обсуждаем is imperfective (from обсуждать).

• Imperfective: обсуждать → мы обсуждаем
  Focus: process, duration, repeated action
  Example idea: We are (in the process of) discussing our graduation day.

• Perfective: обсудить → мы обсудим
  Focus: result, completion
  Example idea: We will (successfully) discuss / have discussed our graduation day.

So if you say:

• Мы обсуждаем наш выпускной день. – we are currently discussing it.
• Мы обсудим наш выпускной день. – we will (at some point) discuss it and finish doing so.

With обсуждать, Russian uses a direct object in the accusative with no preposition.

• обсуждать что? – to discuss what?
  обсуждать план, идею, проект, день (to discuss a plan, an idea, a project, a day)

So:

• Мы обсуждаем наш выпускной день. – literally We discuss our graduation day.

If you use говорить instead, you need a preposition:

• Мы говорим о нашем выпускном дне. – We are talking about our graduation day.

So:

• обсуждать что (no preposition)
• говорить о чём (with о)

Наш выпускной день is in the accusative case, because it is the direct object of the verb обсуждаем.

• Verb: обсуждать кого? что? → accusative
• Direct object: наш выпускной день

However, день is masculine and inanimate, and for such nouns:

• Nominative = Accusative in form.

So:

• Nominative: наш выпускной день
• Accusative: наш выпускной день

They look the same, but the function in the sentence is accusative (object).

Наш agrees with день in gender, number, and case.

• день is masculine singular.
• In the accusative, with a masculine inanimate noun, the form is the same as nominative: наш.

So:

• Masculine: наш день
• Feminine: наша неделя (our week)
• Neuter: наше утро (our morning)
• Plural: наши дни (our days)

Here we have:

• наш (masculine singular)
• выпускной (masculine singular adjective)
• день (masculine singular noun)

All three agree.

Выпускной день literally means graduation day.

Important nuances:

• выпускной день – the day of graduation in general.
• выпускной вечер – the graduation party / prom night (very common phrase).
• Often people just say выпускной and mean the graduation party / prom.

So:

• наш выпускной день – our graduation day (the whole day of finishing school / university).
• наш выпускной вечер – our graduation party / prom in the evening.

Yes, you can say both, but they are not identical:

• Мы обсуждаем наш выпускной день.
  – We are discussing our graduation day
  (more focused, purposeful, often more detailed or planned discussion).

• Мы говорим о нашем выпускном дне.
  – We are talking about our graduation day
  (more general chatting; not necessarily focused or organized).

So обсуждать suggests:

• analyzing
• planning
• going through details

Говорить о suggests:

• just talking, chatting, mentioning.

Обсуждать is neutral in style. It is fine in:

• casual conversation
  Мы обсуждаем фильм. – We are discussing the movie.
• more formal or professional contexts
  Мы обсуждаем проект. – We are discussing the project.

For a more casual, less serious tone, people might also say:

• Мы болтаем о нашем выпускном дне. – We are chatting about our graduation day.
• Мы разговариваем о нашем выпускном дне. – We are talking about our graduation day.

Russian has no articles (no a / an / the).

In this sentence, наш (our) already makes the phrase specific:

• наш выпускной день – our graduation day
  (this already has a definite sense, similar to the in English).

If you removed наш:

• Мы обсуждаем выпускной день. – We are discussing graduation day.
  That could be more general, but still there is no article in Russian.

You can change the word order in Russian; it will still be grammatical, but the focus changes.

Standard, neutral:

• Мы обсуждаем наш выпускной день.
  (focus on the action: we are discussing our graduation day)

Possible with emphasis:

• Наш выпускной день мы обсуждаем.
  Emphasis: our graduation day is what we are discussing (and not something else).

Russian allows flexible word order mainly for emphasis and information structure, not for grammatical reasons. The sentence in your example is neutral and natural as given.

Stress and rough transcription:

• Мы обсуждáем наш выпускнóй день.
  • мы – [mɨ]
  • обсуждáем – [əpsʊʐˈdajɪm] (stress on -да-)
  • наш – [naʂ]
  • выпускнóй – [vɨpʊsˈknoj] (stress on -ной)
  • день – [dʲenʲ]

Main stress points:

• обсуддáем
• выпускнóй

You would make день plural and adjust everything to plural:

• Singular: наш выпускной день – our graduation day
• Plural: наши выпускные дни – our graduation days

Changes:

• наш → наши (plural)
• выпускной → выпускные (plural adjective)
• день → дни (plural noun)

Then the full sentence in plural:

• Мы обсуждаем наши выпускные дни. – We are discussing our graduation days.