Yes, this is a two-stop push.
First, disclaimer: I shoot mostly B&W nowadays. Color has a lot more exposure latitude, though I think it prefers overexposure to underexposure (which you did). (This latitude is why pushing/pulling is less common with color.) Sometimes you can shoot color negative film at the wrong speed, process it normally and still get usable results. As for whether to get your film push-processed or develop it normally -- I'll leave that advice to other who know better and just talk a bit about pushing/pulling.
Pushing the film is underexposing and over-developing. Pulling is overexposing and underdeveloping.
Here's my layperson's understanding of how it works: Exposing silver halide to light causes some of it to convert to metallic silver (the dark bits on your negative). Development pretty much continues this process, with the crystals that got more light exposure getting converted (reduced) more. So if you don't give the film enough light, you can compensate by giving the developer more time to continue this process. (As I understand it, if you exposed film long enough, you could forgo developer altogether and plop it right into the fixer, the chemical that removes the un-converted silver halide.)
My go-to for low-light photography is 400-speed B&W with the camera set to 1600. It's two stops underexposed, so I extend development time to compensate. Results are (hopefully) good negative density, but more grain and contrast than I'd get if the film were shot with the camera set to 400.
Here's an example of HP5 at 1600.
Pushing/pulling w/ B&W is easy. Every B&W has slightly different development times (which varies with any of the zillion developers you can use), so pushing/pulling is easy to do, just develop longer. Most films/developers have a chart showing development time at different ASA settings.
Color negative film is meant to be a standardized process, called C-41. Pop it into the machine and it's all automated. Hence the difficulty with pushing/pulling; I don't know if they change the speed of the machine or change the temperature of the developer, but it's an out-of-the-norm process, so labs charge more.
As I said, color neg film has a lot of latitude (tolerance for improper exposure). Ektar a bit less than some general-use films, perhaps, since it's meant to be pro-level stuff. C41 film was meant for use in, among other things, cheap cameras with fixed exposure settings, so it's designed to return usable results with less-than-optimal exposure. It's really amazing technology and hats off to Kodak and Fuji et al for how well it works.
Your best bet might be to call Darkroom or another lab that does pushing/pulling of color film, tell them what you did and ask them. Yours is not an uncommon mistake; they've probably seen it a zillion times before and will know whether you need to push-process or develop normally. As Xkeas mentioned, pushing/pulling can affect colors, so you might be OK with slightly thin negatives that have good color.
The good news is: This is a mistake one rarely repeats!
HTH
Aaron
PS, technically this is a 35mm equipment thread -- you might get better/more comprehensive answers on the Color: Film, Paper and Chemistry forum. I'm fairly new here so not sure what the policy is on moving posts.