It is implied (heavily) that Hagrid's umbrella contains the pieces of his wand which was broken after his expulsion from Hogwarts. However, there are many examples of wands being broken and they either don't work at all or work very poorly.
Even though Hagrid isn't an excellent magician he is still able to perform simple spells using his umbrella. How is it that his umbrella works so well for him?
All of the broken wands we see in canon were broken quite recently; Hagrid's wand would have been broken about fifty years ago, so he's had a whole lot more practice in trying to use it.
Furthermore, the other broken wands in canon are only poorly mended; as I recall, Harry barely even tries to jury-rig a fix for his wand, and Ron's is Spell-o-Tape'd together. Hagrid's wand, on the other hand, is effectively being held in a splint, if we assume that he put the broken pieces of his wand in the stem of his umbrella instead of in the ribs or something like that.
In fact, one could even argue that by putting the broken pieces of his wand in there, Hagrid has effectively crafted himself a new wand - after all, wands are composed of a magical core surrounded by by a wooden body, and by putting the magical core of his broken wand in the wooden stem of his umbrella Hagrid may have accidentally made something that is significantly more effective than a broken wand, yet not quite as powerful as a "real" wand.
Finally, and perhaps the largest stretch, consider the fact that wands are quasi-sentient - much like many animals, with which Hagrid has quite an affinity. His wand may simply like him enough that it's willing to put in the extra effort to help him out, and thus work better than one would expect a broken wand to.
Thus, although there doesn't seem to be much canon evidence for why he's able to do magic, it is reasonable to think that Hagrid's ability to use magic with a broken wand stems from years of casting spells with it, and his more-effective repairs to it, and perhaps even his sunny disposition.
I have no canon reference to support this but one possible explanation that came to me would be that Dumbledore repaired Hagrid's wand using the Elder Wand. We see in the HP7 that the Elder Wand has the ability to repair a wand that has been broken. Since Dumbledore was the rightful master of the Elder Wand before we meet Hagrid in HP1 it is possible that Dumbledore used this power to repair Hagrid's wand.
A wand is just another way of channeling magic, and that magic is channeled using the core which is encased inside wood, in case of a wand!
It's highly possible that Hagrid's wand's core is in the umbrella, but I really think the core would have been broken. I believe Dumbledore just went on ahead and did a reparo like Harry did, in Deathly Hallows.
In Sorcerer's Stone, Hagrid takes Harry to Olivander to get his wand, but Olivander recognizes Hagrid (Not hard to do) and asks him about his wand.
"Rubeus! Rubeus Hagrid! How nice to see you again.... Oak, sixteen inches, rather bendy, wasn't it." "It was, sir, yes," said Hagrid.
"Good wand, that one. But I suppose they snapped it in half when you got expelled." said Mr. Ollivander, suddenly stern.
"Er -- yes, they did, yes," said Hagrid, shuffling his feet. "I've still got the pieces, though," he added brightly.
"But you don't use them." said Mr. Ollivander sharply.
"Oh, no, sir." said Hagrid quickly. Harry noticed he gripped his pink umbrella very tightly as he spoke.
"Hmmm," said Mr. Ollivander, giving Hagrid a piercing look.
Hagrid lied twice, his wand wasn't in pieces (not anymore) and he still uses it, but only under Dumbledore's permission.
Harry Potter wiki does suggest the possibility that Dumbledore might have repaired Hagrid's wand!
Following his expulsion, the Ministry of Magic forbade Hagrid from practising magic and destroyed his wand. Albus Dumbledore nonetheless convinced then Headmaster, Armando Dippet, to give Hagrid the job of Gamekeeper and allow him to remain a resident of the school. Hagrid continued to do magic using his pink umbrella, which most likely contained the broken fragments of his wand, or perhaps even his entire wand intact, reconstructed for him by Dumbledore using the Elder Wand.
It doesn't actually work that well at all.
If you'll recall, Hagrid's very first act of magic in the entire series was to try to turn Dudley into a pig. And perhaps quite thankfully, he doesn't entirely succeed - instead he merely places a curly pig tail on Dudley's end that has to be surgically removed (yeowch).
So clearly, although it's been mended into the shape of an umbrella, it's not nearly as effective as a proper wand, though at least not as disasterous as Ron's Backfiring wand in Year 2.
It's also worth noting that once Hagrid has been cleared of his criminal charges, he may have gotten it properly repaired or replaced - even if Harry's wasn't capable of being repaired, it's possible Hagrid's, being of a different make and 'rather bendy', could have been repairable, or at least usable for a new wand.
Ollivanders said "I am sorry, very sorry, but a wand that has suffered this degree of damage cannot be repaired by any means that I know of." That is the key, he does not know how to but he is not closed-minded to there being other means to do it or that maybe another person knows how to.
I think Dumbledore fixed Hagrid's wand but much later on when he was older, not immediately when he was made into Hogwart's Gamekeeper. Most likely he fixed his wand when he became a member of the Order of the Phoenix, but because he never finished school and spent a lot of time without being able to practice his magic, this made it difficult for him to use it as fluidly as someone that had more recent practice in the use of his/her magic.
Later on, Hagrid could not practice his magic often because he was too busy with his work in Hogwarts and also because nobody ever thought, of the people that knew about him having his wand repaired, to give him some help in exercising his magic abilities. That's why he could not turn Dudley into a pig and could only give him the tail.
Besides, being a half giant he had an advantage by having a strong tolerance against the curses and jinxes that the Death Eaters threw at him. That compensated for the low magic skills he had.
And well, that is my opinion.
Reading some of the responses above, although very interesting, it seems that people are unwilling to admit that there are some plot holes here. I think it is scraping the barrel to suggest that Ollivander over thought the question and presumed Harry meant good as knew rather than repaired (they were in the middle of a war remember, I think Ollivander would have been clear on the need for Harry to have a wand that is functioning hence why he informed him that Malfoy's wand would now work for him) and the theory that Hagrid has more practice in using a broken wand does not go on anything in the books. I'm sorry but this is just a hole and Rowling herself has admitted that there are some. Although, I admit that Ollivander saying anyway that he knows of does leave some ambiguity, however as Harry finally managed to repair his wand with the Elder Wand that would suggest that is in fact the method he does not know of. I believe that sentence was a prelude to the Elder Wand as a solution. I find it dubious that someone as learned in wandlore as Ollivander would not have come across fixing broken pieces of a wand together in a manner similar to Hagrid using his umbrella. It is of course possible as someone suggested that Dumbledore may have mended Hagrid's wand for him without anyone knowing. After all he was the one that sent Hagrid to fetch Harry from Godric's Hollow and later from the Dursley's to get him ready for Hogwarts and Hagrid says that he was permitted to use magic so it would be logical to presume that Dumbledore must have been aware that Hagrid had means of using magic, i.e. a wand, and that he may have been involved in mending it, probably not telling Hagrid how he'd done it.
I really do believe that Dumbledore repaired it for him with the Elder wand. It's quite possible (even probable) that Hagrid didn't know about the existence of the Elder wand and believed that all wands could mend other wands, which is why he didn't spill the beans about the Elder wand. In my opinion, that's the most logical explanation.
I think there is enough evidence in the canon suggesting that a broken wand is not functional. Hagrid's wand does seem to malfunction. But considering Hagrid is an unqualified wizard (who had at the most 3 years of magical education) who probably wasn't a very powerful wizard in the first place (I would imagine him to have been somewhat like Neville, passionate and good-hearted but magically untalented), I think his poor magic should be attributed to his abilities rather than his wand.
Therefore, a reasonable conclusion is that this is simply one of the non-too-rare inconsistencies between JKR's earlier writings and her later books. However, I actually think there could be another credible explanation apart from a plot hole. I don't know if this is supported by anything JKR might have mentioned elsewhere, but I think there might actually be a minimum length for a wand in order to be functional. Hagrid, being a half-giant, presumably has a wand approximately twice as long as a typical wizard's wand. Assuming the minimum length theory to be valid, Hagrid's wand is probably still around or above the minimum length even if it's broken in half.
