Aneurysms are swollen blood vessels. When left alone, they will continue to swell until they burst, causing hemorrhaging and lowering the patient's vitals. Since the vessels are usually very small, the magnifier is used during the operation.
To treat a Normal Aneurysm, the brown sedative must be injected into it. Then, before the sedative wears off, the aneurysm must be cut out with the scalpel. After which, the aneurysm is to be removed with the forceps. The blood pooling from the vessel is then to be drained, and the forceps are needed to reconnect the vessel. Finally, suture the vessel to cure the aneuryum.
Large Aneurysms appear in New Blood and Under the Knife 2, and are treated the same way as Normal Aneurysms. However, since they are formed at the junction between 3 vessels, a synthetic vessel needs to be positioned between the ends of the blood vessels with the forceps. In Under the Knife 2, these synthetic vessels are already in the correct angle, while in New Blood, these synthetic vessels need to be rotated into place by twisting the Wiimote until in the correct angle. Finally, suture all 3 ends of the vessels to cure the aneurysm. If the vessel is left untreated for too long, blood begins pooling, obscuring view and breaking the chain.
Many players find trouble with aneurysm operations as while they are treating one, other aneurysms may burst. The general idea is to sedate and cut all the aneurysms first, so that they do not pose any threat. Once done cutting all aneurysms in view, remember to scan the area to check for other aneurysms.
Treat and cut the larger aneurysms first. They require more dosage of sedative, and don't cause bleeding (and vital loss) after you've cut them, unlike normal aneurysms. A full syringe of sedative should also be able to fully shrink (or at least sufficiently maintain) multiple swelling normal aneurysms.
As you treat the aneurysms, more may appear outside your view, the nurse would often alert you to this. This often occurs when you've closed a blood vessel before suturing it. If this happens, quickly connect any open vessels and then scan around to find the remaining aneurysms.
Most missions dealing with aneurysms will also launch a final wave of 4 or more aneurysms. This is usually a good time to use the Healing Touch. Otherwise, note that some vessels start out in a greater danger of bursting than others - memorize which these are, and treat those first, if you want the "no aneurysms burst" bonus these missions offer.