This war is unmistakably illegal.
Under international law—at least as defined by the UN Charter, to which the United States is a founding signatory—one country can only legally launch attacks inside another under one of three conditions: if the intervention is authorized by the UN Security Council; if it's a cut-and-dry case of self-defense; or if assistance is requested by the other country's government.
It's true that in Iraq at least, the government requested U.S. assistance in stemming the spread of IS—an intervention promoted in Washington as part of an effort to prevent the genocide of Iraqi religious minorities like the Yazidis (remember them?). Yet the United States has continued launching strikes on IS positions in Iraq long after the crisis on Mt. Sinjar was putatively resolved.
But in Syria, not a single one of these conditions applies.
In a letter to the United Nations explaining its strikes on Syria, the Obama administration claimed that it had the right to attack IS positions that the Syrian regime was "unable or unwilling" to eradicate itself. IS, the administration argues, has used its strategic depth in Syria—where no U.S. intervention has been formally invited by the still-sovereign Assad regime—to attack Iraq, which has requested U.S. assistance.
Here it almost seems like the U.S. and Iraqi governments are taking a page from IS itself and attempting to erase the Iraqi-Syrian border. It's true that IS is a big problem on both sides of the Iraqi-Syrian border, but the government of Iraq simply has no legal authority to direct a third country to attack Syria. (Imagine a hypothetical scenario in which Russia attacks the United States because Syria requested help in warding off foreign intervention in its territory. This won't happen, but it shows the inane implications of the administration's rationale.)
Additionally, any claims the White House makes about "self-defense" at this stage are spurious, since U.S. intelligence agencies have confirmed that IS presently poses no threat to the U.S. homeland. This makes sense—after all, who has time for international terrorism when you're also trying to conquer and govern new territory? No need to attack the "far enemy" when your objectives are achievable where you're already fighting. (Unless, of course, the far enemy suddenly starts bombing you.)