I'm not entirely sure what you might be meaning by the word "era", so I'll give a few interpretations.  

Shakespeare wrote most of his plays right around the year 1600.  1589 to 1613 to be exact.  That puts Shakespeare writing during both the reign of Queen Elizabeth I and King James.  

If that's not what you are looking for, then I'll go for a literary genre.  The dates that Shakespeare was writing in coincide with the Renaissance.  Depending on the source of information, the Renaissance dates can shift around a tiny bit, but no matter what, Shakespeare was writing his stuff during the Renaissance.  

Lastly, "era" might be referring to the type of English being written in.  Shakespeare wrote in Early Modern English.  That always surprises students to learn, because they all swear that he must be writing in Old English.  Then I show students what the Old English of Beowulf looks like.  It can't be read.  It doesn't even look like English.  Middle English is readable, but still tough.  A good example of Middle English would be Chaucer's writings.  Shakespeare is using an early version of the same English that we use today—Modern English.  

In addition to the previous answers, Shakespeare's language is what we call 'Early Modern English'. The English Language has gone through many centuries of evolution and it is common to call Shakespeare's language, 'Old English', when in fact this is not true. Old English was the language spoken by the Anglo Saxon conquerors of the British Isles and it is close to Old Icelandic. This was the language in which Beowulf was written. It is very foreign to modern English and quite unrecognisable.

Norman conquest of England in 1066 began what we call 'Middle English'. This was the language of Chaucer and is much more familiar to us today than Old English - though still quite different.

Then we come to the English Renaissance and a linguistic change known as The Great Vowel Shift that completed around the mid 16th century. After this and into the late 16th to the 17th centuries, we have writers such as Donne and Milton, and later Shakespeare and Marlowe writing in Early Modern English.

The changes that shaped the English Language following the Industrial Revolution marks the point where we see genuine modern English that is quite familiar to us today. Jane Austen, Charles Dickens, George Elliot etc.

So Shakespeare's 'Era' is a complicated topic! Since Shakespeare is a literary figure, it would be relevant to consider his 'era' from a linguistic perspective too.

I hope that helps!

I guess this depends some on what you mean by "era."  In terms of simple dates, Shakespeare wrote most of his plays between 1589 and 1613.

If you are talking about what literary era he was writing in, people tend to say that he was part of the English Renaissance.

If you are asking about historical eras, most people associate Shakespeare with Elizabethan England.  Queen Elizabeth I ruled from 1558 to 1603 and was, therefore, the queen during the first half or so of Shakespeare's career.  King James succeeded Queen Elizabeth and was ruling at the time Shakespeare died.