In the plays of Shakespeare we see two ways of writing employed: blank verse, which is written in iambic pentameters (a 10 syllable line with 5 "feet", with the first syllable unstressed and the second stressed") and then prose, which has no specific order or structure. What is interesting to note is who speaks prose and who speaks verse. Normally, the noble characters who make up the main plot speak in verse, whereas the lowly, more humble characters who make up the subplot speak in prose, however, some characters will change from verse to prose or vice versa, and it is worth thinking about why this is so and what Shakespeare is trying to communicate through this change.
An example of blank verse would be Portia's speech on mercy in The Merchant of Venice:
The quality of mercy is not strained.
It dropeth as the gentle rain from heaven
Upon the place beneath.
Note how each of the first two lines have 10 syllables (we pronounce "heaven" with one syllable for the purposes of this line) and it has a set rhythm. However, note how in the same play, Shylock and Tubal in Act III scene 1 speak in prose together. I hope this helps you distinguish between the two!
The first answer gives you excellent tips (especially about why specific characters rely on verse while others use prose). Here's an easy way to determine if someone is speaking in verse versus prose. If you are reading a Shakespearean play that uses most of his original language (not a No Fear Shakespeare or abridged version), simply look at the lines. If a character's lines are uneven, with some being much longer than others, the lines are in verse. When the character's lines look like a standard paragraph with words going from margin to margin, they are in prose.
Similarly, if you read lines written in verse out loud, you should hear a distinct sing-song rhythm (that's the iambic pentameter). If you read lines out loud, and they don't seem to have a beat to them, they are most likely written in prose. Hope this helps!