Your Internet provider may have speed tests available on their website, so checking for such tests that are specialized to your connection might be a good idea. Also, depending on your operating system, there may be some sort of speed test or meter already installed on your machine for your use.
Your best bet is to use one of many “speed tests” available online. These tests will check the speed of your LAN Ethernet connection, most of them for free, providing quick, easy, accurate results.
There are two different types of speed tests available: ones that are done in-browser, and ones that are completed with downloaded software. Typically, how they work is by testing how long it takes to transmit data between your Internet connection and the server of the site.
Either method works, but have their particular advantages and disadvantages. While in-browser speedtests tend to be more convenient, they don't provide nearly the statistical depth or flexibility of a downloaded speedtest software.
For some examples: Speedtest.net offers easy, free, global Internet speedtests in-browser, and LAN Speed Test 1.0 is available free for use and download.
If you're not inclined towards any of these, you can also use a script to check for the speed yourself. This article provides both a review of Speedtest.net and a way of testing your LAN network speed using a simple PHP script.