First question: the mishna at the beginning of tractate Kiddushin makes it clear that the only way a Jewish married woman is allowed to marry someone else is either with a Get (a Jewish divorce), or if the first husband dies.
Deuteronomy 24:1 makes it clear that the first husband has to give her a writ of divorce. (A civil divorce is issued by the courts; a Get, on the other hand, is a document that he hands her, stating "you are no longer married to me.")
It is not uncommon for a woman who's not all that religiously observant to have had a civil divorce; now she affiliates more with Jewish practice (or meets someone who does), and has to call up her ex-husband and ask if he's willing to go through a Jewish divorce ritual (i.e. order the rabbis to write up a Get and give it to her). I know of several such cases firsthand.
Note that as Judaism does not acknowledge any binding tie of marriage between a Jew and a non-Jew, if the first husband wasn't Jewish, this would not be necessary.
If any of this may be practical for you or anyone you know, please contact the experts: info@bethDin.org. They are based in New York, but can make connections or references all over the world. Or if you're in Europe, try the London Beth Din. Here's a good article from them about the process; their phone number is listed as +44 020 8343 6270.
Second question: does a civil marriage still require a Jewish divorce ritual to dissolve?
The short answer is that this was subject to a lot of debate in the twentieth century. (It may even depend on how seriously people took civil marriage; in the Soviet Union, it was argued, people got them and dissolved them all the time, they just needed the bigger apartment! An even weaker case came up in the US if the civil marriage was a sham for the sake of someone's immigrant visa.) Everyone agrees that when at all possible, she should contact the ex and go through a Jewish divorce ritual. (If he's no longer alive, or he wasn't Jewish, then it's not a problem.) There are responsa of Rabbi Moshe Feinstein (who allowed it if necessary), his senior peer Rabbi Joseph Elijah Henkin (who didn't), and many others on this subject.
Similarly, here's the London Beth Din's policy (emphasis added):
Since Jewish law regards the mere fact of setting up home and living together as being possible evidence of marriage, the question should be asked of the Beth Din whether a Get is required in the event of ... registry office marriages