When Voldemort came to kill the Potter family James and Lily realize he is there with a little bit of time to spare. James provides a momentary buffer and Lily is in the room with Harry. Why did she not grab Harry and apparate away from Voldemort?
Besides for the debatable reason of not having a wand as mentioned in the accepted answer, there is a more basic reason why she could not have Apparated. In Deathly Hallows when the trio went from Shell Cottage to Gringotts, we have the following description:
All three of them glanced back at Shell Cottage, lying dark and silent under the fading stars, then turned and began to walk toward the point, just beyond the boundary wall, where the Fidelius Charm stopped working and they would be able to Disapparate.
So we see that you cannot Disapparate from within the confines of a Fidelius Charm. As the Potters house in Godric's Hollow was similarly protected by the Fidelius Charm, it would have been impossible for Lilly to Disapparate out of it.
Furthermore, even without the Fidelius Charm it is likely that she would not have been able to Disapparate out of the house. In the beginning of Half-Blood Prince we have the following exchange between Harry and Dumbledore:
"Professor, why couldn’t we just Apparate directly into your old colleague’s house?"
"Because it would be quite as rude as kicking down the front door," said Dumbledore. "Courtesy dictates that we offer fellow wizards the opportunity of denying us entry. In any case, most Wizarding dwellings are magically protected from unwanted Apparators. At Hogwarts, for instance —"
"— you can't Apparate anywhere inside the buildings or grounds," said Harry quickly. "Hermione Granger told me."
Thus, it is quite possible, even likely, that the Potter's house had an anti-Apparition charm placed on it for protection. Therefore, the reason why Lilly couldn't Disapparate out is the same reason why Voldmeort couldn't Apparate in — it wasn't possible.
You can not easily (if at all) Apparate without a wand (HP Wikia has a long list of examples and reasoning explaining why). Side-along with an infant would be even less possible.
In DH, Voldemort recollects that neither James nor Lily had their wands when he killed them.
He was over the threshold as James came sprinting into the hall. It was easy, too easy, he had not even picked up his wand... “Lily, take Harry and go! It’s him! Go! Run! I’ll hold him off!” Hold him off, without a wand in his hand!... He laughed before casting the curse... “Avada Kedavra!”
He climbed the steps, listening with faint amusement to her attempts to barricade herself in... She had no wand upon her either...
So Lily could not Apparate due to lack of wand.
I'm going to say it's because she didn't have her wand. There is some debate as to whether a wand is strictly required, but I think it is. Plus, Slytherincess already covered the "Gryffindors are brave" angle.
Their house was under the Fidelius Charm. I am not sure if the Fidelius Charm itself comes with this additional bonus but almost all occurances seen in the series of the Fidelius Charm also come with the inability to apparate into or out of the protected area. This would prevent Lily from grabbing Harry and apparating out of the house. There is evidence that you are able to apparate within the confines of the protected area so it might have been possible for her to grab Harry then apparate to the edge of the protected area, run away and then apparate to a different location. However, this all happened so quick there was probably no chance to think of such a plan quickly and Voldemort probably would have struck fast enough to prevent such a plan.
It does strike me as a little foolish not to carry your wand with you when you know you're in mortal danger. The main reason for her not to simply apparate away, however, might have been that, like someone above state, Voldemort was too quick. If Lily had tried to grab Harry, he would have finished her off in that moment.
It states in Deathly Hallows when Ron and Harry were prisoners in the cellar of Malfoy Manor, "Ron was now trying to Disapparate without a wand," This shows that this was a foolish act and that it was fruitless.
Well, Lily is a Gryffindor. It doesn't really seem her style to run and leave loved ones (James) behind. Voldemort gave Lily a choice to live or die ("Step aside, silly girl") and Lily chose to die to protect Harry. Harry was in his cot when Voldemort entered his nursery and it may have been that by the time Lily took him out of his cot, Voldemort would have killed her. But regardless of all that, I keep coming back to the fact that Lily was a Gryffindor and would have never bowed to death.