Zombies attack in numerous ways for the chance to kill or infect individuals.
This is without a doubt the most dangerous and common attack. Bites are notorious for getting a person infected. Most viruses seem to be transmitted by saliva if a person gets infected by a bite since there is almost no other way (except for blood) that a person can be infected by such contact. Bites can also cause nerve paralysis, as the teeth could hit a nerve. It is best to put a bitten person out of their misery, or if you can and the person has been bitten very recently, you may have a chance to quickly amputate the affected body part (like an arm or leg) and get the person professional medical attention as soon as possible to avoid death by blood loss.
This is a zombie's common secondary attack. There is a great chance for an infection to start due to this, the scratches can cause wounds which can be prone to infection, as claws may have infected blood or other bodily fluids on them.
Faster zombies can run and grab or ram people to incapacitate them, giving the zombies a better chance of killing or infecting the person.
It has been proven, that when a human dies, everything their body would normally do during life stops. When you work out, for example, you're tearing your muscles on the molecular level so that they can heal and grow bigger and stronger. If a zombie tears its muscle, it cannot heal so the tissue, therefore, remains torn forever. A zombie most likely will not be able to run simply because it doesn't have the cognitive ability to. Thus the reason for the rumored shambling shuffle is notorious to the common zombie.
The faster zombies can pounce on an individual and claw that their abdomen, giving a person large wounds and quite possibly killing him/her. It would be difficult to shove a pouncer off of you as zombies may have abnormal strength and the unbearable pain the chest and abdomen can give fatigue to the arms and legs. It is always best to have a buddy nearby to kill any pouncers. A notable example of a pouncer is the Hunter from Left 4 Dead. But this attack type is unlikely due to the fact that muscles would waste away quickly after death. the Hunter
Some of the stronger zombies (body builders, wrestlers, etc.) may try to punch their prey as instinct compels them to. This attack, however, should be fairly harmless but can cause momentary disorientation. This is also unlikly due to the fact the virus eliminates all thoughts other than feeding.
Some infected, such as those who have been in the water for a long time, may swell up until its organs became filled with water, causing it to have a bloated belly. If distance is kept, these will pose no real threat.
Zombies may also vomit as a sign of sickness and may blind or infect those who are vomited on. A notable example would be the "Infected" from the 28 Days Later franchise. Some infected have the ability to launch their projectile bodily fluids half the length of a football field, so be wary of who you choose to attack at nearly any range.
This is more of a psychological attack than physical, zombie moans can cause an extreme drop of morale over prolonged periods of time. Zombies do not sleep, and if they are able, they moan non-stop, day and night. While barricaded in a base, this constant reminder of their presence can easily drive people insane. It has been reported that people have even killed each other to occupy themselves instead of hearing the zombie moans nearby. A sound-proof base will be a must if you plan to stay there long term.
More intelligent kinds of undead still retain the ability to use weapons, ranging from simple hand tools and rocks to more complicated things like firearms and explosives. Notable examples of armed zombies are those who infected by Las Plagas in Resident Evil 4, which are armed with multitude of weapons ranging from pitchforks to miniguns, and Zombines (zombified Combine soldiers) from Half-Life 2: Episode One, which are still armed with their hand grenades, but use them for kamikaze attacks instead of throwing them. This attack strategy is next to impossible due to the fact the virus eradicates all thoughts other than to eat.