In the Fellowship of the Ring when the Fellowship was entering Lothlorien Frodo hears Galadriel's voice and he seems bewildered by it. So how did Galadriel know his name? is it because of her Elvish senses or something?
Well, firstly, he didn't just hear voices in the book.
All the Elves in Lothlorien knew something of the Fellowship, from the messengers that Elrond had sent out before they departed:
‘But we have heard rumours of your coming, for the messengers of Elrond passed by Lórien on their way home up the Dimrill Stair. We had not heard of - hobbits, or halflings, for many a long year, and did not know that any yet dwelt in Middle-earth. You do not look evil! And since you come with an Elf of our kindred, we are willing to befriend you, as Elrond asked; though it is not our custom to lead strangers through our land.
Setting that aside, Galadriel knew the name and kind of every member of the Fellowship:
‘Also,’ said Haldir, ‘they bring me a message from the Lord and Lady of the Galadhrim. You are all to walk free, even the dwarf Gimli. It seems that the Lady knows who and what is each member of your Company. New messages have come from Rivendell perhaps.’
Haldir credits this to further communication between Rivendell and Lothlorien. Celeborn seems to confirm this:
When all the guests were seated before his chair the Lord looked at them again. ‘Here there are eight,’ he said. ‘Nine were to set out: so said the messages. But maybe there has been some change of counsel that we have not heard. Elrond is far away, and darkness gathers between us, and all this year the shadows have grown longer.’
(LotR, The Mirror of Galadriel)
On top of that, Galadriel has a particularly refined telepathic ability:
‘What did you blush for, Sam?’ said Pippin. ‘You soon broke down. Anyone would have thought you had a guilty conscience. I hope it was nothing worse than a wicked plot to steal one of my blankets.’
‘I never thought no such thing,’ answered Sam, in no mood for jest. ‘If you want to know, I felt as if I hadn't got nothing on, and I didn't like it. She seemed to be looking inside me and asking me what I would do if she gave me the chance of flying back home to the Shire to a nice little hole with - with a bit of garden of my own.’
That's funny,’ said Merry. ‘Almost exactly what I felt myself; only, only well, I don't think I'll say any more,’ he ended lamely.
All of them, it seemed, had fared alike: each had felt that he was offered a choice between a shadow full of fear that lay ahead, and something that he greatly desired: clear before his mind it lay, and to get it he had only to turn aside from the road and leave the Quest and the war against Sauron to others.
‘And it seemed to me, too,’ said Gimli, ‘that my choice would remain secret and known only to myself.’
‘To me it seemed exceedingly strange,’ said Boromir. ‘Maybe it was only a test, and she thought to read our thoughts for her own good purpose; but almost I should have said that she was tempting us, and offering what she pretended to have the power to give. It need not be said that I refused to listen. The Men of Minas Tirith are true to their word.’ But what he thought that the Lady had offered him Boromir did not tell.
(LotR, The Mirror of Galadriel)
‘I know what it was that you last saw,’ she said; ‘for that is also in my mind. Do not be afraid! But do not think that only by singing amid the trees, nor even by the slender arrows of elven-bows, is this land of Lothlórien maintained and defended against its Enemy. I say to you, Frodo, that even as I speak to you, I perceive the Dark Lord and know his mind, or all of his mind that concerns the Elves. And he gropes ever to see me and my thought. But still the door is closed! ’
(LotR, Mirror of Galadriel)
Galadriel and Frodo, both as Ringbearers, also apparently have a particularly powerful bond:
Yet even so, as Ring-bearer and as one that has borne it on finger and seen that which is hidden, your sight is grown keener. You have perceived my thought more clearly than many that are accounted wise. You saw the Eye of him that holds the Seven and the Nine. And did you not see and recognize the ring upon my finger? Did you see my ring?’ she asked turning again to Sam.
‘No, Lady,’ he answered. ‘To tell you the truth, I wondered what you were talking about.
(LotR, Mirror of Galadriel)
Elrond's sons Elladan and Elrohir are referenced as making a journey to a strange land beyond the Dimrill Dale; this is after the Council of Elrond where Frodo was chosen, therefore it is logical to suppose they told Galadriel what was occurring and who was who.
As Galadriel is in possession of her mirror that shows
things that were, and things that are, and things that yet may be
It would be a fairly safe assumption she could have gained this information from this.
Galadriel also knew something had befallen Gandalf as she could no longer see him from afar.
"Nay, there was no change of counsel," said the Lady Galadriel, speaking for the first time. Her voice was clear and musical, but deeper than woman’s wont.
"Gandalf the Grey set out with the Company, but he did not pass the borders of this land.
Now tell us where he is; for I much desired to speak with him again. But I cannot see him from afar, unless he comes within the fences of Lothlórien: a grey mist is about him, and the ways of his feet and of his mind are hidden from me."
Showing that she is indeed able to scry upon the fellowship
If she was able to do this she would have known he wasn't alone and would likely have known of his companions.
Elrond had also informed his kin in Lothlorien of the Fellowship including the make up.
"Here there are eight," he said.
"Nine were to set out: so said the messages. But maybe there has been some change of counsel that we have not heard. Elrond is far away, and darkness gathers between us, and all this year the shadows have grown longer."