John Parkhurst, in his Hebrew lexicon entry for תחש, points out that the ancient translations all agree that it is not an animal but a color:
Before the Tabernacle was constructed, God told Moses and the elders to come worship Him:
1 Now He said to Moses, “Come up to the Lord, you and Aaron, Nadab and Abihu, and seventy of the elders of Israel, and worship from afar.”
9 Then Moses went up, also Aaron, Nadab, and Abihu, and seventy of the elders of Israel, 10 and they saw the God of Israel. And there was under His feet as it were a paved work of sapphire stone, and it was like the very heavens in its clarity. -Exodus 24:1, 9-10 (NKJV)
The pavement under God's feet was like a sapphire, which is deep blue in color, or, as Michael Paul Heart says in his answer, indigo (between blue and violet):
Exodus 25 begins with Moses receiving instructions on what offerings the Israelites are to bring for the construction of the Tabernacle:
1 Then the Lord spoke to Moses, saying: 2 “Speak to the children of Israel, that they bring Me an offering. From everyone who gives it willingly with his heart you shall take My offering. 3 And this is the offering which you shall take from them: gold, silver, and bronze; 4 blue, purple, and scarlet thread, fine linen, and goats’ hair; 5 ram skins dyed red, badger skins [tachash skins], and acacia wood; 6 oil for the light, and spices for the anointing oil and for the sweet incense; 7 onyx stones, and stones to be set in the ephod and in the breastplate. 8 And let them make Me a sanctuary, that I may dwell among them. 9 According to all that I show you, that is, the pattern of the tabernacle and the pattern of all its furnishings, just so you shall make it.” -Exodus 25:1-9 (NKJV)
Not only do the ancient translations have "blue skins" instead of "badger skins" in verse 5, Josephus also mentions that among the acceptable offerings were skins dyed either blue or scarlet:
HEREUPON the Israelites rejoiced at what they had seen and heard of their conductor, and were not wanting in diligence according to their ability; for they brought silver, and gold, and brass, and of the best sorts of wood, and such as would not at all decay by putrefaction; camels' hair also, and sheep- skins, some of them dyed of a blue color, and some of a scarlet.... -Antiquities of the Jews, Book 3, Chapter 6, Section 1
The Tabernacle is referred to in the Psalms as God's footstool:
1 The Lord reigns;
Let the peoples tremble!
He dwells between the cherubim;
Let the earth be moved!
5 Exalt the Lord our God,
And worship at His footstool—
He is holy. -Psalm 99:1, 5 (NKJV)
6 Behold, we heard of it in Ephrathah;
We found it in the fields of the woods.
7 Let us go into His tabernacle;
Let us worship at His footstool. -Psalm 132:6-7 (NKJV)
and as such the outer covering of "tachash skins" would appropriately be of a blue color, much like the color of the sapphire pavement Moses and the elders saw beneath God's feet in Exodus 24.
Badgers, porpoises, sea cows, and seals are all unclean animals. In order to get the skin of any of these animals you would obviously have to kill it first. However, touching the carcass of an unclean animal is specifically called a sin and required a trespass offering:
1‘If a person sins in hearing the utterance of an oath, and is a witness, whether he has seen or known of the matter—if he does not tell it, he bears guilt.
2 ‘Or if a person touches any unclean thing, whether it is the carcass of an unclean beast, or the carcass of unclean livestock, or the carcass of unclean creeping things, and he is unaware of it, he also shall be unclean and guilty. [...]
5 ‘And it shall be, when he is guilty in any of these matters, that he shall confess that he has sinned in that thing; 6 and he shall bring his trespass offering to the Lord for his sin which he has committed, a female from the flock, a lamb or a kid of the goats as a sin offering. So the priest shall make atonement for him concerning his sin.’ -Leviticus 5:1-6 (NKJV)
It doesn't make much sense that God would command His people not to touch the carcasses of unclean animals, while at the same time requiring skins of unclean animals for the outer covering of the Tabernacle which itself was constructed as the means for atoning for sins. This is perhaps in part why some of the more modern Bible translations such as the English Standard Version (ESV) and Good News Translation (GNT) tend to go with "goatskins" or "fine leather," respectively, in addition to the Egyptian word for leather as Jon Ericson mentions in his answer.
The New American Standard Bible (NASB) goes with "porpoise" because of the very similar sounding Arabic word دُخَس (du-has), as Jon Ericson also mentions in his answer. Gesenius talks about this in his lexicon entry for תַּחַשׁ but goes on to say that it "has a wider extent, and also comprehends seals, which in many respects resemble the badger, and which were of frequent occurrence in the peninsula of Sinai...." This is most likely why the American Standard Version (ASV) went with sealskins, as seals would be much easier to trap or catch than dolphins/porpoises, and would also be more numerous, thus providing greater amounts of skins that could be sewn together.
"Sea cows" is basically a compromise between seals and dolphins, as they are of a larger size than seals and also inhabit the waters of the Red Sea.
As for "badger," the NKJV retains what the KJV translated for "tachash," as Jon Ericson points out in his answer. The KJV translators most likely yielded to the authority of the Rabbi's and Talmudists when they decided to go with "badger." See Gesenius under תַּחַשׁ where he says he has no hesitation in acceding to the opinion of the Talmudists who take it to be an animal such as the badger or weasel.
Personally, I am quite convinced that the ancient translations were correct in that "tachash" was not an animal but a color (deep blue, like a sapphire). The Septuagint (LXX), Aquila, Symmachus, Theodotion, Jerome, and Josephus are, in my opinion, of greater authority than the Talmudists, especially when combined with the fact that badgers, porpoises, sea cows, and seals are all unclean.