Finding replacement lightbulbs can be difficult if the the bulb to be replaced I's discontinued. If I find a bulb with the same dimensions, Kelvin and lumen, does it matter if the wattage differs? For example, what if the old bulb was 10W and the new one is 7W?
TL;DR Lumens = amount of light, Kelvin = color of light, CRI = quality of light, Watts = power used
There are a number of different measurements of a light source. Note that I said light source, not light bulb. That's because with modern long-lasting LED lighting, often the best thing to do when an old bulb dies is to actually replace the entire fixture. Replacing the entire fixture costs a bit more and is definitely more work, but if the fixture, with light source, lasts 10 years or more then that really isn't much of a concern.
A lumen is a measure of actual light. A lumen is a particular amount of light, or illumination. Lux and foot-candles are measures of how much light is on a particular area. In other words, a particular light source will produce a certain number of lumens. But your goal, when picking lighting for a new location, is to provide enough lumens to light up the area - which is really lux. But when you are replacing a light source, using the previous number of lumens is a good way to do things, possibly increasing or decreasing based on your perception of the amount of light with the old light source. Keep in mind that many types of light sources will gradually decrease the amount of light they produce over their lifetime, so a little extra light at the beginning may really translate into a longer-lasting usable light source.
Kelvin is a temperature scale, applied to light. It gets complicated, but essentially comes down to a particular number will translate into a particular color of light. While most light sources we use in residential and commercial applications are basically "white", they actually vary quite a bit depending on the technology (incandescent, fluorescent, LED, etc.) and variations within each technology. The terminology can get very confusing. Many people aren't too particular about color temperature, though it can be noticeable if you have a chandelier with bulbs of different color temperatures. I don't think any computer-generated illustration really does this concept justice due to the limitations of computer monitors, so I recommend going to a hardware store, big box store or lighting store that has a display of different color temperature light sources side-by-side.
The Color Rendering Index, or CRI gets rather technical, but it essentially tells you how good the light source is at producing light across the full spectrum, which translates into colors looking the way they should, with natural daylight as the standard. Incandescent bulbs are normally 100. Typical baseline (except for real junk) is 80, but high-quality LED and fluorescent lights can get into the 90s. If you ever wondered why the lighting in big warehouses or street lamps can be bright enough to see well yet colors seem "wrong", that is because they use sodium or mercury lamps with awful CRI.
Watts are a measure of power, particularly electricity. More watts = more power = more cost = more heat. Any power that isn't turned into light is turned into heat, which is OK (but not ideal) in the winter and double trouble in the summer, and absolutely useless almost all the time for outdoor lighting.
A light source (incandescent) that uses 60 W to produce 800 lumens will produce the same amount of light as a light source that uses 13 W (compact fluorescent) or 8 W (LED). Wattage tells you nothing about the quality or intensity of the light. However, with 40 W, 60 W, 75 W, 100 W bulbs the standard for decades, fixtures have been designed around 'x bulbs @ y watts', and we have conditioned ourselves to think "a small bedroom needs 3 60 W bulbs worth of light". When in reality it is "a small bedroom needs 2,400 lumens worth of lights". The result has been the "60 W equivalent" game, which actually provides reasonable guidance for replacement of existing bulbs (whether with LED bulbs or replacing the fixture altogether) but simply confuses things when it comes to new installations.
In general, especially with partially or totally enclosed fixtures (whether recessed cans or covered bulbs), fixtures are designed to safely dissipate a specific amount of heat. More than that is a problem. An old fixture designed for 60 W incandescent bulbs can handle 13 W fluorescent or 8 W LED. And it can even handle a much brighter (more lumens) 15 W LED. But the other way around is a problem. If you have a fixture designed for 8 W LEDs, it might have a label "maximum 10 W". If you put a 60 W incandescent bulb in to that fixture you could easily start a fire, unless it was really designed for higher heat dissipation and just labeled that way by default (management decreeing that all fixtures should be "LED only" to look good environmentally, even if the fixtures haven't changed from old incandescent designs).
On the other hand, FreeMan has pointed out some situations where an outdoor light using LEDs might not produce enough heat to keep it clear of ice and snow. That is more of a problem with vertical lights (e.g., traffic lights) than downward facing lights and will vary a lot depending on weather conditions and other factors.
Wattage really has no bearing on how good a light is. But two light sources with the same specifications (lumens, Kelvin, CRI) and different wattage have different efficiency. In most, but not all, cases, higher efficiency wins as it will cost less for power over the lifetime of the light source.