Their shape seems to have varied. A lot. This is perhaps most obvious if we look at their women. We have, on the one hand, desireable women such as Gerðr, whom Frey falls in love with and sends Skirnir to woo, as told in Skírnirsmál, and who is described as so fair that she lighted sea and sky. Also, Freyäs father, Njord, was married to the giantess Skadi, who in Grímnismál is described as the "fair bride of the Gods". Several of the Gods has also been known to couple with giantesses and produce offspring; in fact, most of the Aesir have a giantess for a mother (the one exeption of which I am sure is Balder).
At the other extreme, we have giantesses such as the (paternal) grandmother of Tyr, described in Hymiskviða as "ugly" and having "nine hundred heads".
Having several heads seems like something of a common trait; one of the first giants, an unnamed figure that is created out of a union between Ymir's feet, had according to Vafþrúðnismál six heads (no other of the poems which mentions him notes this detail).
That there is no easy way to tell if a person is a giant or not is best seen in the story of how Sleipnir came to be: An unnamed figure offers the newly established gods to build them fortifications in three years with the aid of no one but his horse. If he manages, he will get Freya, the Sun and the Moon. He almost does, but Loki manages to lure away his horse in the guise of a mare, and so he fails. Only then does the Aesir realise that the builder was a jötunn and calls for Thor, who smashes his head. (Gylfaginning, chapter 42).
However, for the most parts, the stories makes it explicitly clear from the start who is a giant and who is not.