The vital equipment for a day's hiking in the mountains will mostly depend on the mountains in question and on the time of the year. The absolute essentials for any full day's hiking in any mountains include food, drink, map and compass, rain gear, matches/lighter and a knife or multi-purpose tool. These are, obviously, in addition to appropriate clothing and walking/hiking boots.
This basic list should be supplemented by items appropriate to the area and season of the hike. John Shannon of the Texas Sierra Club developed the original "ten essentials" proposed in the 1930s by the Seattle Mountaineers into the "ten essential groups"s, which form a great starting point for choosing the vital equipment for a day hike.
(1) Clothing. Depending on temperature, sun and other conditions (not the least of which are the presence of biting insects), choose comfortable, light and breathable clothing. For extreme hiking, man made fibres that wick moisture are recommended, but more moderate walks leave more leeway for your preferences. In addition to sports fabrics, high-tech wool or wool mixtures, bamboo and silk are all luxurious and efficient. Layers are always preferable. Weather in the mountains can change very rapidly, and what was a bright, hot and sunny day might turn into a storm, with a pouring rain and a temperature dropping by ten degrees. Long trousers are better than shorts unless it's very hot, and will protect you from insects. Always bring an item of rain gear unless you are in the rare arid mountains where rain doesn't happen. Spare socks don't take much space and can make all the difference, especially in wet conditions.
(2) Boots. Probably the most essential of the essentials, a pair of hiking (in the UK: walking) boots is crucial to the enjoyment of the hike and might be crucial to the safety. And by boots, boots are meant. You might get away in some hills with trekking sandals, or even trainers, but in most areas you will need proper and well fitted boots.
(3) Food. Essential for a full day's walk and recommended for anything but a short amble within a mile of a pub or a mountain hostel. Bring packed lunch as a minimum (after a decent breakfast at home) and additional snacks. Even if going for a shorter walk, still bring nutritious, high energy snacks. Chocolate biscuits and sport bars, dried fruit and nut trail mix (or your favorite fruits and nuts), fruit cake, bananas are all good for energy that you might need. For very intensive hikes and strenuous climbs high-fat foods like nuts or chocolate might not be the best idea: carbs rule, complex for sustenance, simple sugar for a quick energy boost. But on most day hikes you will be fine with any nutritious and light to carry food you like.
(4) Water. This is even more important than food,and obviously the more important, the hotter and drier the weather. In cold weather, a flask of a hot drink (tea, coffee, or even soup) may be the best thing you take on a day mountain hike. If the weather is hot, taking some means to purify water is the best way to save weight you will need to carry.
(5) Fire. Although you are not likely to need fire – and in many areas fires are either not advised or explicitly forbidden – it is worth taking fire making tools, as in an emergency a fire may save your health and life and increase chances of being found when lost. Matches, lighter or a fresnel lens and possibly something that will be kept dry for tinder or/and a small cube of solid camping fuel don't take much space but might come in very useful.
(6) Map and compass. An absolute essential even in a busy area with well marked trails: know how to read the map and know how to use a compass. A GPS might be useful too, especially where there is fewer managed trails.
(7) Light. A torch (flashlight in the US) is useful if there is any chance that you are going to be caught by the darkness. This means that you probably don't need it for a 6 hour hike in the Scottish mountains in the summer where it stays light for 20 hours, but if you are planning a full day hike there is always a risk that something might keep you. As with many items, it's a compromise between weight and risk. A small wind-up (those you grip are better then those with a crank) torch might be the best solution.
(8) Sun protection. The actual need for that will depend on the weather, season, latitude, altitude and a presence of snow. Sun reflected on snow can cause snow blindness and burn skin quickly, so a wide-brimmed hat, sunglasses and even sunscreen are vital in many locations.
(9) Medical supplies. A full first aid kit per each person is not necessary for a day hike, but between you the groups should have some basic medical supplies, including elastic bandage, plasters (band aids in the US), aspirin (painkiller and blood thinner). Some people carry more extensive supplies and anybody with health issues should take anything they might require, as well as carry emergency information on their person (or make sure their hiking companion know their medical history). First aid course and ability to do CPR is probably the best thing you can have (and doesn't weigh anything). Insect repellent is extremely useful in midge, mosquito, blackfly or tick infested areas.
(10) A fully charge up mobile phone is a good idea, though you won't have coverage I many places. Always leave information on where you are going with a family member, friend, at a hostel or at a mountain rescue station (whichever appropriate). A whistle, signal mirror or, for extreme hikes, even a personal locator beacon might be useful.
(11) Tool, or tools: a multi-function knife like a Swiss Army knife or another good, reliable, light knife is also an item that at least one of the hiking party should carry.
Depending on the location, season, individual needs and the distance, altitude and terrain the selection of vital items will change, but it's likely that every day hike will require items from each of the above groups.