Until when during the day may one put on Tefillin?
One is not to don Tefillin after Shekiah/sunset. [See Q&A!]
If one did not put on Tefillin the entire day, may he do so after sunset, before Tzeis Hakochavim?
Yes. However one is to put the Tefillin on without a blessing.
May one put on Tefillin by Mivtzaim after sunset?
Yes, one may put Tefillin on Jews up until nightfall. Some rule that one may be lenient to do even with a blessing.
If one is already wearing Tefillin must he remove them when night enters?
Past Sunset: If one is wearing Tefillin before sunset he does not need to remove the Tefillin when sunset arrives.
Past Tzeis Hakochavim: When nightfall arrives [Tzeis Hakochavim] one is to remove his Tefillin if he is in a public area. If however one is in private, in an area without any other peple, he does not need to remove the Tefillin even after nightfall. Likewise if one is in a Beis Midrash, even if there are other people around, he may continue wearing them.
Erev Shabbos and Erev Yom Tov: On Erev Shabbos and Erev Yom Tov one must remove the Tefillin prior to sunset.
 Siddur Admur; Implication of Beis Yosef 30; opinion in P”M 30 A”A 2
Ruling of Admur in Shulchan Aruch and other Poskim: In the Shulchan Aruch 30/2 [see also 18/8 regarding Tzitzis] Admur rules that one may don Tefillin until nightfall/Tzeis Hakochavim. So rules also the M”A 30/2; Mordechai [brought in Beis Yosef ibid]
 The reason: Perhaps Admur in the Siddur follows the ruling of the Rambam that Biblically prohibits wearing Tefillin after sunset. [See Rambam 4/10]
 P”M 30 A”A 6; M”B 30/2 and 5
Does this apply even according to Admur in the Siddur? Some write that the ruling of Admur in the Siddur does not apply to one who did not yet don Tefillin that day. [Rav Ginzberg in Hiskashrus 783; Author of Piskeiy Teshuvos brought in Hiskashrus 807] Others argue that perhaps according to Admur in the Siddur doing so is forbidden. [Rav Raskin in Miluim on Siddur]
 M”B ibid
 Poskim of previous Question; This applies even in accordance to those who follow the ruling of Admur in the Siddur: Rav Ginzberg in Hiskashrus 783/787/800/807; Author of Piskeiy Teshuvos brought in Hiskashrus 807,
 Author of Piskeiy Teshuvos ibid
 As many Poskim rule Tefillin may be worn even initially until nightfall with a blessing, and so rules Admur in 30/2. Hence why be stringent on them like the ruling of the Siddur. [ibid]
 Siddur Admur
 The reason: From the letter of the law, it is only forbidden to don the Tefillin at night, however if one had them on prior to nightfall [Tzeis Hakochavim] he may continue wearing them past nightfall. Nevertheless, we do not rule this way practically [Halacha Vein Morin Kein] due to worry that one may come to initially don them at night. Thus one should not continue wearing them in the presence of other people so people don’t come to think it is permitted to wear Tefillin at night. [ibid]
 The reason: As in a Beis Midrash the people are knowledgeable and aware that you cannot initially put Tefillin on at night. [ibid]
 Siddur Admur
Other opinions: Please see http://www.dailyhalacha.com/displayRead.asp?readID=1520 that according to Chacham O. Yosef zt”l one may recite a blessing on tefillin until 18 minutes past sunset.
However, the Rambam in Hilchos Tefillin, Chapter 4, Halacha 11 says explicitly that a person who puts on tefillin after sunset transgresses a Scriptural prohibition.
It appears to me, in my humble opinion, that one who did not wear tefillin the entire day and the sun has already set is to wear them up until 72 minutes past sunset, as according to Rabbeinu Tam and the SA it is still day [or possibly day] until that time. Vetzarich Iyun gadol if one may be lenient to wear tefillin past sunset [even a moment past sunset] on Friday or erev Yom Tov if he did not wear them before. It appears logical one may not, and so paskened to me HaRav Shlomo Miller, a Litvishe posek in Toronto. But many are lenient to daven Mincha past sunset on Friday, in particular Chassidim who follow the opinion of Rabbeinu Tam, and one may transgress a shvus during bein hashemashos. Tzarich iyun.