Arianespace is their biggest competitor, they are the de facto rulers of the commercial launch market, with SpaceX hot on their heels. The Ariane 5 launches twin payloads about 7 times a year, for a total of about 14 total spacecraft launches per year. SpaceX is about at a similar cadence but has yet to hit the overall schedule and mission reliability figures that Arianespace has achieved so it's still a bit of a race. Launching on Ariane is the easy, safe choice. Launching on the Falcon 9 is still considered to be a little bit risky, more so from a schedule perspective (which is still a big deal in the industry).
SpaceX is increasing its launch cadence and reducing its prices substantially (through reusability) so they are becoming more competitive by leaps and bounds every moment. If they can follow through on their cadence increases (right now they are around an average of 3 weeks between launches and a minimum of 2 weeks) and cost reductions (see recent flight) no one else will be able to keep up with them. Their internal cost reductions on reuse are unknown but their current price reductions are around 30%, which is off of their already incredibly aggressive low launch prices. If SpaceX can provide launches at around $45 mil and can launch often enough to satisfy demand reasonably then they'll end up with the majority of the commercial launch business eventually. If you can put your satellite in orbit reliably and save on the order of $50 million then you'd be insane not to do so. There's a bit of an insurance premium with launching on SpaceX currently but that should even out after a while and the huge cost reductions swamp that difference regardless (even if, for example, SpaceX were to blow up another rocket tomorrow).
The only realistic competitor for SpaceX in the near term is Blue Origin, once they develop their orbital launcher which will also have VTVL first stage reusability. The timeline on that is sometime before 2020, but that's likely to be the soonest any real competition for SpaceX's business model shows up. Once it becomes commonplace for launches to involve first stage reuse (which will probably happen over the next 5 years or so) I'd expect a lot of the other rocket companies (like Boeing and LockMart) to come out with similar designs (or go out of business or be relegated to national payloads only). It's a lot easier to put forward some hypothetical "better" design for your next generation rocket when everything is on paper, it's a lot harder to pretend your rocket is still better when you're getting your face pushed in by the competition on price by huge margins on a regular basis.
Once SpaceX has the Falcon Heavy running (which will be even more reusable since the upper stage is a smaller proportion of the rocket) they'll be able to launch nearly anything anyone else can, and at dramatically lower costs. Once they're able to launch GPS satellites (which they've already got orders for) as well as the big national security birds (on the FH) for tens or hundreds of millions of dollars less than ULA then almost all of that business is going to flow to them and LockMart/Boeing are going to have some tough times.