The radioactive, complex mutagenic enzymes in the spider's blood that were transferred at the time of the bite triggered a profound and significant change in the genetic expression of Peter Parker's entire body. The enzymes contained the codes for certain spider traits that were inserted into Peter's DNA. It transformed the way his body systems work, including his glands, nerves, and muscles. Spider-Man possesses superhuman strength, reflexes, and equilibrium; the ability to cause parts of his body to stick with great tenacity to most surfaces; and a subconscious premonitional "danger" sense.
Peter's physical structure has been altered and enhanced to superhuman proportions. His overall metabolic efficiency has been increased, allowing him to fuel superhuman muscles, far stronger and denser than any normal human being. The origins of Spider-Man's strength is in the webs. Spider webbing is a protein-based fiber, some types of which are stronger than steel. Human muscles are also protein-based fibers so, by chance the rewriting of Parker's DNA led to his already present muscle protein structure being rewritten to be more like spider webbing so his human muscles gained titanic strength. His body also now produces more of the contractile proteins of muscles, actin and myosin. More contractile protein, further his enhances his musculature strength. With his musculature altered as it is, Spider-Man is capable of lifting over ten tons under optimal conditions and at maximum exertion, in both a leg press or a bench press.
The make-up of his skeleton has also been augmented to support these weights, so that he doesn't snap in half when lifting a car over his head. With this degree of strength, Spider-Man can easily punch through thin layers of carbon steel, sustaining only minimal damage to his tissues—roughly similar to a normal person punching through a sheetrock wall. His stamina has also been augmented to slightly enhanced levels, however, on a proportional level, he is roughly same to the average person, lifting 300 lbs is extremely difficult, and an act difficult to sustain. Spider-Man is no different, except that his minimum is roughly 10 tons, his maximum while in stressed 'fight or flight' conditions is around 130 tons. This is because he's been able to lift a pile of debris, that was estimated to weigh as much as a locomotive. His strength is more than sufficient enough to propel himself to impressive leaps and bounds or feats of overall agility, especially when combined with his insubstantial mass.
Speed is directly related to the strength of muscle. As good a shape as he's in, a powerless Spider-Man can bench a fair bit more than his own 165 lb weight, he could bench 200 lbs without his powers. His powers make him approximately (at the very least)100 times stronger. And increasing the energy/strength available increases the speed by the square root of that figure. Therefore his powers only make him 10 times faster. Which given even the average for his physical acuity, puts him at approximately 100 miles per hour for short sprints. But we must take into account that spiders can run 70 body lengths a second. Peter is 5'10/1.76m so he should have a maximum sprint speed of 275mph. He can run at over 45 mph quite a while at his peak ability.
Spider-Man has reflexes 15 times faster than normal humans. He has enhanced energy resources for his strength and those higher energies manifest all the way down to his metabolic intracellular processes, the nerve conduction velocity of his nervous system is faster simply by sheer brute energy resources. Spider-Man is able to respond to stimulus in a virtually instantaneous manner. He is frequently able to dodge automatic gunfire from multiple sources provided he has some distance between himself and the source, and is able to move with enough room to generate necessary momentum.
Spider-Man is capable of incredible feats of superhuman agility. Agility is not so much an attribute as it is a combination of attributes, many of which Spider-Man has at vast levels. His super-strength and speed allow him to do the necessary movements. The fact that his "web based" tissues have much higher elasticity despite their strength gives him amazing flexibility. His superhuman reflexes help him adapt to any changing environment with vast rapidity. They also allow him to find equilibrium almost instantly in any given situation. This enhanced nerve efficiency is also applied to his cerebellum, allowing him superhuman coordination and aiding his balance. Also an increase in proprioreceptors and an increase in their sensitivity aids his coordination and balance even more so. His stamina with movement is similarly enhanced, and he can keep himself moving quickly for nearly an entire day. This agility is augmented and enhanced by the spider-sense, rendering him even more capable of feats of miraculous evasion. All these things combined give Spider-Man super-agility.
Spider-Man's entire body has been made somewhat more resilient. His web-like tissues are very tough, especially since they're woven into the meshes of his body. This gives him a great deal of shock absorbing capability and general resilience. In effect, his skin and muscles seem 'thicker', and his bones are more resistant to breaking. Nothing akin to Kevlar, however a needle will have a bit of trouble breaking the skin, for example, but a sword will be more than sufficient for the job. Overall, his body is simply hardier. His bones are stronger, partly to support the tremendous weights he is capable of lifting. His skin is tougher, to reduce the impact of blows, but a powerful punch can still leave him reeling, even from one who is merely an exceptionally strong human. He's far from invulnerable, but it's enough to take the edge of off many injuries enough to keep him going, or save him from the absolute brink of death on many occasions.
The sheer hardiness and efficiency of Spider-Man's body grants enhanced health, he tends to heal at an accelerated rate, and he's seemingly resilient to poison and disease. His aging is slightly retarded as well. Bruises and small cuts vanish within a few days. Broken bones also heal faster, probably 2 to 3 times faster than an average human, as do larger wounds. However, his rate of healing is not so dramatic as to not require such things as disinfectant, bandages, stitches, and the like. Gunshots are still gunshots, for example, and even one is going to take Spider-Man out of action for some time a number of days, perhaps over a week. Spider-Man's accelerated healing is not quite dramatic enough to make most people question the speed in which Peter's cuts and bruises vanish, unless they were extremely careful to keep tabs on his status.
Spider-Man's exposure to the mutated spider venom induced a mutagenic, cerebellum-wide alteration of his engrams resulting in the ability to mentally control the flux of inter-atomic attraction (electrostatic force) between molecular boundary layers, allowing Spider-Man to form temporary molecular bonds between his body and another object as he manipulates the inter-atomic binding forces of matter. This enables Spider-Man to cling to solid surfaces with vast force and scale walls simply by placing his hands on them.
The most incredible of all of Spider-Man's many superhuman attributes is his 'spidey-sense' or 'spider-sense' that warns him of impending danger through a tingling sensation in the back of his skull and upper neck. The secrets of Spider-Man's spider-sense lie within three factors, electro/magnetoreception, extreme sensitivity to atmospheric vibrations, and super-fast reflexes. Parker's entire dermis acts as a sensor that uses ambient energy to see through objects, almost like omni-directional x-ray vision. Since this is too much information for the normal human consciousness to process, his subconscious does the work and only alerts the conscious mind to things of importance.
Spider-Man can't be snuck up on because of the omni-directional "sight" of his sense, it always alerts him which direction danger comes from. Weapons (guns, knives, brass knuckles, etc.) have their own distinct reflective signatures from ambient energy due to their shapes and compositions, so spider-sense would pick them up within range and warn Spider-Man, whether they were hidden or not.
Everyone also has their own unique biochemically generated electromagnetic aura. Spider-Man's subconscious can use the spider-sense to see the EM aura of a super-villain in costume and remember it, and then if it happens to encounter the same aura surrounding a civilian, it will warn Spider-Man that the person was the villain out of costume by simply giving him "a bad feeling" about that person. This also allows Parker to spot people (villains) in disguise. It also allows him to sense people in hiding.
Spider-sense also acts as radar due to its omni-directional nature, allowing Spider-Man to sense his surroundings even when he's blinded or in total darkness, because there's always some form of radiation present. Spider-sense warns Spider-Man if he's about to step on a creaky floorboard when he's trying to be quiet. This works because his subconscious knows he doesn't want to make noise and is scanning for any way he might, this includes sensing if the structural integrity of the board he's about to step on isn't as sound as the one he just quietly stepped on.
Spider-Man knows when he's being watched too, so he doesn't accidentally change in front of someone he doesn't want to know his secret. With spider-sense constantly "x-ray viewing" in all directions, it knows when he's being looked at because it can see the looker and warns him accordingly. This same reason is how he knows he's being aimed at with a weapon, his sense is literally "looking down the barrel".
When disarming a bomb, his subconscious using his spider-sense scans the entire design of the bomb, including which wires current is flowing through and using Parker's affinity for science, figures out which wires to cut, then alerts him accordingly. Its omni x-raying abilities also warn him of what he would consider a potential danger even on the other side of a wall or door.
His spider-sense is also able to alert him to the hostile attitudes of people due to the changes in EM aura that occur with changes in emotional states. A supreme dermal sensitivity to vibrations aids the EM aspect, allowing movement to be felt from quite a distance. This combined with his ultra-fast reflexes kicking in instinctively, makes his spider-sense possible.