Our tags are actually a pretty good guide here.
The period of history roughly from the 15th century to the mid 20th century
Contemporary history describes the period timeframe that is without any intervening time closely connected to the present day and is a certain perspective of modern history.
These are the categories used by historians.
Now of course people like to propose their own "Age"s to slot the contemporary period (and usually all or part of the modern period). The only one you are probably safe with is "Modern Age", which is essentially a synonym for "Modern Period" (usually with Contemporary thrown in).
Anything else you hear generally comes with a specific outlook or theory attached. Of those, ones I have heard are:
- Renaissance - My Art history teacher in particular insisted we are still in the Renaissance.
- Industrial Age - The idea behind this is that we are in a culture roughly shared with people born after the advent of steam power and the industrialization that came in its wake in the 1760's.
- Information Age - This one takes the Industrial Age idea, but claims that that advent of global communications and computing since the late 20'th Century puts the Contemporary period itself in a whole different class. As a computer person myself, I'm kind of partial to this one.
(Discourse on the last bullet follows. Skip it if you like)
Douglas S. Robertson took the idea of the Information Age and went even further. He classifies all societies based on the amount of information, in bits, that a typical member has access to. I believe this is called "Informationalist History".
h is the amount of info one mind can hold, and is probably in the vicinity of 5Mb (5*106 bits).
- Level 0 - 107 bits (h) - Pre-Language
- Level 1 - 109 bits - Language
- Level 2 - 1011 bits - Writing
- Level 3 - 1017 bits - Printing
- Level 4 - 1025(?) bits - Computers
The exponent on that number of bits is the important thing. How far one society outclasses another can be gauged by the difference in those exponents. This is why Native Americans, the most advanced of whom barely had writing, had no hope of competing with Europeans with printing presses, but under the right conditions could actually replace a society of Europeans with no printing press a few years earlier. Being a couple of orders of magnitude back can perhaps be dealt with. However, be several back and you're lucky if they bother to treat you as the same species.
An Informationalist would say we are in the Computer Age, and that further human progress to any new level is going to require us to find ways around our current limitations on information access (particularly combing through massive amounts of it in new and more productive ways)