Is it possible for a woman to conceive from two different men and give birth to half-siblings?
Yes, this is possible through something called heteropaternal superfecundation (see below for further explanation).
Of all twin births, 30% are identical and 70% are non-identical (fraternal) twins.
Identical twins result when a zygote (one egg, or ovum, fertilized by one sperm) splits at an early stage to become twins. Because the genetic material is essentially the same, they resemble each other closely.
Typically during ovulation only one ovum is released to be fertilized by one sperm. However, sometimes a woman's ovaries release two ova. Each must be fertilized by a separate sperm cell. If she has intercourse with two different men, the two ova can be fertilized by sperm from different sexual partners. The term for this event is heteropaternal superfecundation (HS): twins who have the same mother, but two different fathers.
This has been proven in paternity suits (in which there will be a bias selecting for possible infidelity) involving fraternal twins, where genetic testing must be done on each child. The frequency of heteropaternal superfecundation in this group was found (in one study) to be 2.4%. As the study's authors state, "Inferences about the frequency of HS in other populations should be drawn with caution."
A search on Google will show you that there have been documented cases where twins had different fathers, including at least one case where a paternity lawsuit follows and the mother admitted to her husband that she had an affair.
There are also cases where twins look very different although they have the same parents. Including cases where one twin looks white and one twin looks black. This can happen when you have one white and one black parent, the children can end up being quite white or quite black and twins can look different. And it can happen if both parents are mixed race; they can have children that are lighter or darker than both parents, and twins can be one lighter and one darker.
Twins looking different by coincidence are much more common than twins with different fathers; as mentioned earlier in cases where there was a paternity lawsuit, only 2.4% turned out to have different fathers. That number will of course depend on the behaviour of people.
One newspaper article http://www.theguardian.com/lifeandstyle/2011/aug/19/my-twins-have-different-fathers describes a case with two days between sex with the two fathers which is more than I thought possible. As an example of dissimilar twins see http://www.babycenter.com/0_strange-but-true-mixed-race-twins-one-black-one-white_10364936.bc showing a family who actually has two sets of twins with one quite light and one quite dark.