Acute means that the condition had a "sudden" or "rapid" onset - common examples are cuts, broken bones, strains, sprains, etc. This is in opposition to an injury or condition that is "chronic" which means something that is long lasting (like epilepsy, hepatitis, or diabetes - see: http://en.wikipedia.org/wiki/Chronic_(medicine) ).
Causation Issues: With accident and negligence cases, you need to show that the acute injury was caused by the incident in question. With acute injuries, this usually is not that difficult if you notice the injury right away, and seek medical treatment to diagnose and document the injury.
Typically, if an injury is considered acute, it means that the injury just recently occurred. Many acute injuries speak for themselves - like if you have an a car accident, hit your head against the side door frame, and cut part of your face or head. It is easy to see what caused that injury. Common sense tells us that if the person was not bleeding prior to the collision, and then after the collision their face is bleeding, that the accident caused the injury. Broken bones, back injuries, neck injuries, injuries to the extremities (like arms or legs) are often easy to confirm that they were caused by the traumatic event in question (the automobile accident for instance). If the injury, disability, and pain was not present prior to the accident, and then it is there after the accident, we don't need an expert to tell us that this was caused by the accident.
Difficult Cases: Some acute cases are more difficult to show "causation". In other words, it may seem to you as the injured person that it was obviously caused in the accident or fall, but an insurance adjuster may be able to create doubt about whether the injury being claimed was caused by the accident.
Injuries that don't Speak for Themselves: Injuries that do not "speak for themselves" are more difficult. In these cases, you need a doctor to explain the injury to a jury, and how that injury could have been caused by the trauma in question. One example is brain injuries. Patients often look normal with brain injuries. In fact, they might even look good from a juror's perspective. These cases need experts / doctors to come in and show the testing that was done, to show how they verified that there was a brain injury, and how severe the injury is.
Gaps in Treatment: In injury cases, you are not dealing with a "health insurer". You are dealing with a liability insurance company. Liability insurers are "occurrence based" claims - meaning the pay claims that are related to a specific "occurrence". If you wait too long after the injury occurs to go in to a medical facility and get it diagnosed, then you give the liability insurance company an excuse to deny your claim.
The liability insurer will ask "how do we know this injury occurred from our incident?" They will ask, "How do we know that you didn't get injured some time after the accident? Maybe working out, or playing basketball with your friends, or working in the yard, or some other reason. Now you are trying to relate your injury back to our auto insurance and get us to pay your claim that is not related to our car accident."
By waiting too long to be seen by a doctor, you have given the insurance company an excuse to deny your injury claim. Similarly, if you go to a doctor, then don't go back for several months, even if your injury never goes away, you give the insurance company ammunition to deny your claim. Again, they will claim that they "find it difficult to believe" that you could be so hurt, yet not go for any medical treatment for several months.
Slow Developing Injury: These are similar to gaps in treatment. If you wait too long to get an injury diagnosed, you give the insurance company an excuse to deny that your medical bills are related to the accident in question. Examples of slow developing injuries are injuries that are masked by more painful, but less serious injuries. For instance, you may break your forearm and have severe pain in that arm for a week or two. But in 6-8 weeks your arm is healed and the pain is gone. About 4-5 weeks into your healing, you start to notice that your neck is hurting. The neck pain was not as noticeable at first, because your arm was hurting so bad at first. It took about 4 weeks for you to really realize that it was more than soreness, but was a real injury. At 6 weeks, you go to the doctor to get your neck checked out. He sends you for an MRI, and finds you have a herniated disc in your neck. The problem here is the "gap". The insurance company will definitely try to claim that this neck injury is something that was either preexisting, or something that happened "after" their accident - because you were never diagnosed with it until 6 weeks after the accident - despite seeing medical professionals. Now, you need neck surgery to repair your injury, which is much more serious than your broken arm was - but you have put yourself into a position where it will be harder to prove the neck injury is related to the accident in question.
Always report whatever your injuries are to your medical professionals, even if the injury is not the most painful one you are currently experiencing - make sure they know of all your symptoms. This is for documentation of the fact that there was an issue that you reported shortly after the incident in question.