MT = multithreaded, most games today only use a few threads heavily, leaving 8-core AMD CPUs mostly un-used. It's commonly believed that directx12 will improve the situation, but early numbers are showing similar improvements on both Intel and AMD CPUs.
HT = hyperthreading. A Core i3 is a dual core CPU, but hyperthreading lets it process 4 threads at once. It isn't as fast as a true quad core, but Intel cores are in the range of 50-60% more powerful than AMD cores, and in applications that only use a couple of cores, Intel has the advantage. However, if you can *fully utilize* all of your cores (as is often the case in things like video encoding), an AMD 8 core CPU that costs $100-130 will match or even slightly pass an Intel i5, which is $190+.
IPC = instructions per clock. Intel cores do more work per clock than AMD CPUs, but AMD CPUs tend to have more cores. Because of this, an Intel quad core almost always beats an AMD 6 core, and in cases where you can't fully use all 8 cores, generally beats AMD 8 core CPUs too.
Quicksync is an Intel feature that lets their integrated GPU encode your videos, and is much faster than relying on the CPU. However, it has limited customization options, so if you're not happy with the quality or size of the result, you might prefer to use your CPU instead. Most people are pretty content with Quicksync's balance between quality, size, and time needed to complete an encode.
~
As has been said, CPU progress has slowed much more in recent years than GPU progress. Someone who bought an Intel i5 in 2011 still has a viable processor, while a mid-range video card bought in the same era will struggle. If looking to build a long-lasting system, a good case can be made for spending relatively more on a processor, since it will last a lot longer. However, you also want to enjoy your games NOW, so a balance must be found.
Generally speaking, an Intel i5 is the minimum CPU needed for a no-compromises experience, with AMD FX-8xxx and Intel i3's being very usable but somewhat compromised cheaper alternatives. People have been saying for years that FX chips will pull ahead of Intel chips because software will become more able to utilize more cores, but this hasn't really materialized. There are a handful of games where the FX-8 performs like an i5 or better, but in most, an i3 and an FX-8 are around the same, and cost around the same, so you get what you pay for.
I don't feel the G3258 is a good option, but it is exceptionally cheap. I'd probably opt for an Athlon 860K in that price range. Some games perform very poorly on the Pentium because it can only process 2 threads at once, which causes stuttering for some reason, despite those threads being processed 50-100% faster than an AMD chip can run 4 at once (slower).
As for the power supply, the trend has been in recent years that PCs are using less and less power. An i3 draws around 50w under full load, a mid-range video card will be 125-175w, and a high-end video card will be in the 250-300w range. The rest of the system is generally peanuts, but you want to have a healthy safety margin so 400w is appropriate for an i3 system. AMD chips are generally more power hungry, with their 8 cores drawing 150-200w (or more for the FX-9xxx chips) when fully utilized. However, most games don't fully utilize AMD chips, so real-world power consumption is often less. 500-600w is more appropriate for an AMD system, but this means more up-front cost, cutting into your budget. Overclocking increases power consumption, but as long as it's a modest overclock, you probably won't need a larger power supply. It's also true that Intel CPUs do more work using the same power, and if electricity is very expensive in your country, you might save a few tens of dollars per year if you heavily use your system by going with an Intel CPU.
~
With regards to cooling, the stock cooler on both Intel and AMD systems is adequate for running at stock speeds, and even for a minor overclock. If you plan to add much voltage, you'll want an aftermarket cooler, but this adds to your total costs. Additionally, you need to spend more on a motherboard to overclock an AMD CPU because they need additional power circuitry.
~
I feel Frozentundra is right in that both an i3 and FX-8xxx are budget-appropriate chips that will serve you pretty well in the foreseeable future. Both have limitations, but are also somewhat less expensive than an i5, freeing up budget for a video card.
An FX-8 is power hungry, but has excellent performance on-tap in programs that can utilize all of its cores. It also has the ability to overclock.
An i3 runs very cool and draws very little power, allowing for a smaller power supply and preventing the need of an aftermarket cooler, and performs admirably compared to a stock FX-8.