This is another slightly contentious issue with some folks holding very firm views on what is 'best'. As with anything, what's best is what suits you the best.
Footwear choice is very important as its main purpose is to provide a firm, supportive platform for your foot and ankle as well as protecting your foot stones, thorns, mud and water.
Footwear for the outdoors can range from open sandals to full-on 4-season mountaineering boots. Most people who get outdoors regularly have a variety of shoes and boots to suit the conditions. I won't talk about sandals except to say that these need not be the leather things you see on holiday in hot countries, In the UK they are likely to be much more robust velcro-strapped affairs meant for anything up to trail walking in warm weather.
Beyond this, the lowest level of 'proper' footwear is probably 'approach shoes'. These are available in a dizzying array of sizes, styles, colours and materials. Sometimes they are general-use, sometimes more specific i.e. with very 'sticky' soft rubber which is perfect for scrambling on rock but not so durable. What you choose will depend on your intended use, pure stylistic preference as well as fit; note, and this applies to all shoes and boots, every manufacturer makes their boots a slightly different shape and even the same size, between two manufacturers, can be an entirely different fit. This is not so much of a problem with approach shoes as the looser fit allows for more wriggle-room but, with the more solid fit of a boot, the wrong shape, fit or size might lead to major problems. Therefore, I would strongly advise against ever buying boots without first trying them on - pretty much every one of the main retailers will allow you take the boots home and wear them indoors for a while to ensure they're a good fit. If you wear them outside, it will show, and they won't take them back.
Approach shoes can be worn for fairly significant days out; for some people its all they wear, apart from in snowy conditions (and even then there's bound to be someone who wears them in those conditions as well). If you don't walk regularly in the hills in this type of footwear, I would caution against wearing approach shoes for more significant days out in the hills; the softer, more flexible soles give a much less solid base meaning your foot will twist more and the muscles will have to work much harder resulting in very tired and sore feet at the end of the day - best to build up gradually, as with anything.
Beyond this, boots (proper) will be referred to as 2-, 3-, or 4-season boots.
2-season boots are typically very lightweight boots, rarely waterproof, intended for use in late spring, summer or early autumn i.e. warmer, dryer conditions. They generally have soft uppers and fairly flexible sole.
3-season boots are designed to deal with wet, muddy conditions although they can also be worn in warm weather as well; you might just find your feet get a bit warmer and sweatier. These boots are generally much more solid and robust although, again, there are a variety of styles from lighter fabric and suede boots right through to one-piece leather boots. They will often have a less flexible sole and may even have a 'shank' within the sole to provide stiffness, depending on intended use (I'll discuss crampon-use further down).
Water-proofing in these boots is most commonly through the use of a waterproof membrane within the boot structure. At the other end of the scale are one-piece leather boots where the waterproof layer is the outer leather itself. These boots tend to be heavier but more robust; it only takes a small piece of grit to get into a membrane boot to cause significant damage, plus the membranes get gradually worn away through use. Leather boots, of course, require ore care in how you dry them and in ensuring they are provided with a good water-proofing wax. For some, myself included, the act of sitting massaging the wax into a leather boot is strangely calming and therapeutic. A note on boot care; never put leather boots on or in front of a radiator or fire to dry. This will cause the leather to dry on the outside causing it to become dry and brittle, and it will soon split. They should be left to dry slowly, at room temperature if you like, so that the leather dries from inside and out.
4-season boots; the big daddies! These are solid, heavy boots with relatively inflexible soles, intended for use in serious mountaineering where you may find yourself climbing steep or vertical ice or using skis. For these activities, a solid boot is preferable as you want the boot to be flexing, bending and popping out of your crampon or ski bindings.
As an aside, let's briefly discuss crampons. These are basically steel frames with spikes that attach to the bottom of the boot to prevent slipping on compacted snow or ice. They tend to be rated C1 to C3:
C1 crampons are generally the most flexible, possibly even articulated in the middle, with 8 or 10 spikes, intended for walking on relatively flat ground. If looking for boots to match, they'll generally be 3-season boots rated as B1.
C2 crampons are intended for walking up inclines and steeper ground but not on vertical ice and, as such, maybe slightly more rigid with 10 to 12 spikes for better grip. They may attach by straps or by ski-type snap bindings. The matching boots would be rated B2 - you can wear C1 crampons with B2 boots but NOT the other way round - C2 crampons on B1 boots is a dangerous combination as the boots can flex and come out of the bindings - generally this happens at the least favourable moment such as on a steep, slippery slope. Trying to put crampons (back) on, on sloping ground, is at best, difficult and, at worst, downright dangerous.
C3 crampons are intended for ice-climbing and are, therefore, absolutely rigid and with additional heavy duty (sometimes removable). They're neither easy or enjoyable to walk in for any great distance.
A side note on sole construction; if you're going to be walking in muddy conditions then it makes sense you'll want a deep tread.
It might also be worth making a note on Vibram soles; there is something of a myth that Vibram is the best and is a must-have for boots. Actually, no; it depends on what you're gong to be doing. Vibram is a triumph of marketing over substance. They are not the best for all conditions; they are very durable, for that they deserve all credit, and are ideal for general walking boots to be worn a lot. But they achieve this by being made of a very hard compound; I have found this to be lethal on damp rocks. Speak to the retailer, tell them what you're planning on doing and don't believe all the hype