Search by date
In the search bar, you can type the date in a YYYY/DD/MM format to filter out emails before a certain date. If you type before:2014/01/1, you'll see a list of all of the emails you received prior to Jan. 1, 2014.
You can also search by how old emails are. If you type older_than:1y, you'll receive emails older than 1 year. You can use m for months or d for days, as well.
If you want to delete them all, click the Check all box, then click "Select all conversations that match this search," followed by the Delete button.
Labels and Categories
If you want to get more specific than a date or time, you can refine your search by looking through the categories the emails are kept in or the labels you assigned them to.
If you have a label called "XYZ," you can search for label:XYZ to find those emails. If you want emails that Gmail has classified as being in your social or promotions folders, try category:social or category:updates.
Now you can take what you've learned and combine them. If you type older_than:1y label:XYZ, you'll see all emails older than 1 year with the label you assigned. For those who have organized inboxes, this is an efficient way to search through and then delete old emails you never use anymore.
If you're looking to get even more advanced, Google offers a full list of search operators to filter through your email and find what you no longer need, including how large messages are, whether they have stars and who sent the email in the first place.