To analyse means to look at the smaller part and show what it adds to the whole. So if you are going to analyse a scene you need to look at its significance to the play as a whole.
For starters, give a VERY brief synopsis of the scene and look at whether it is a major turning point for the action of the play. Another thing to look for is what thematic ideas it contains or continues from the rest of the play.
Characterisation is a vital thing to look at and this means deciding what characters are significant in the scene and what their actions add to the plot and themes of the play.
Symbols and language devices are also important in Shakespeare's plays. In order to analyse a scene it is important to find what symbols or literary devices have been used and look at what they represent.
The final main area to look at would be setting. Look at the physical setting of the scene and look at how important it is to the play.
William Shakespeare needs no real introduction as his name is a world-wide phenomenon. One of the benefits of reading Shakespeare is that his plays are still relevant even to a twenty-first century reader. This is because his themes and story-lines have a universal quality and are timeless and appropriate. He talks about love in all its complexity and broaches controversial topics of race discrimination and gender roles, for example with his tragedies, such as Othello, and comedies, such as Twelfth Night being equally merited. Whilst The Tempest is categorized as a comedy it also falls into the romance category due to its serious undertones.
To analyse a scene from any Shakespeare play, you need to place your scene in context in terms of the play itself and to have an understanding of the language used which can be confusing to a modern reader. It is a good idea to use a version of the play that includes notes which remove the potential for misunderstanding or for missed significance. You can then interpret the intention or importance of something and its relevance to the overall picture with a clearer understanding.
Shakespeare likes to leave the reader (or audience) with a degree of uncertainty in the interpretation so that that reader can draw his or her own conclusions. This allows the reader to ask and answer his own questions and ensures that he examines motives and circumstances and does not make assumptions which many Shakespeare characters do and around which the plot often revolves as miscommunication and misunderstanding is responsible for the downfall of many of Shakespeare's heroes. In Othello, Othello's assumption that "honest Iago" is in fact honest leads to his acceptance of the "ocular proof" (III.iii.364) presented to him and after which he kills his beloved Desdemona.
In analyzing any scene of Shakespeare, give some attention to the soliloquies as they often expose contradictions, hints and they can reveal a lot about a character or theme. In Othello, the reader learns a lot about Iago which Othello is not privy to and the themes are also revealed in much of what Iago says. Consider the theme of appearance versus reality in this play which you would discuss in an analysis of Act I, scene i where Iago makes it quite clear that "I am not what I am" (65) in his soliloquy from line 41 to 65. The fact that Iago is unconcerned that Roderigo hears this statement about his duplicity speaks to Iago's character and arrogance.
Therefore, in an analysis of a Shakespeare play
1. Consider the relevance to a twenty-first century reader.
2. Discuss whether it is a tragedy or comedy and what makes it so.
3. Briefly outline what precedes your scene so that the scene has context.
3. Clear up any difficulties with the language in the scene of your choice so that the person reading your analysis has a clear understanding of your interpretation.
4. Discuss the themes, setting, characters which appear in your chosen scene and how the message is conveyed through the dialogue, actions and so on.
5. Further to point 4, explain the contribution that this scene makes to the overall plot or which theme it supports or which characters it most develops.