A video card (Graphics card) is the most important component of a computer that indicates the overall gaming performance of a PC. Same as CPU, a video card has a dedicated chipset called "GPU" (Graphics Processor Unit) which is responsible to receive geometry information from the CPU as an input and renders the data to deliver the final picture to monitor as an output. A GPU is made of different processing units which are grouped together and work in parallel to deliver the final image to your monitor. The performance of a graphics card depends on many factors such as GPU's architecture, clock and efficiency, memory bus, bandwidth, capacity and clock, number of shader (Stream processing) and texture units(TMUs), Render Output Pipelines (ROPs) and game engines.
GPU's Clock and Efficiency:
The architecture of the GPU is referred to physical design and mathematical optimization of GPU that greatly affects the performance of a GPU in certain areas. GPU's core clock is measured in Megahertz (MHZ). As a rule of thumb, higher clock rates increases the performance in games. The efficiency of GPU is a good indicator of the technology used to build that GPU which is referred to transistor size and is measured by nanometer (nm). Smaller size produce less heat at higher core clocks and consume less power.
Shader (SPUs) and Texture Units (TMUs):
Shader units (Stream processors) are responsible to render Vertex, Hull, Domain, Geometry, Pixel and Compute shaders. Texture units in a GPU are used to filter and add texture data to the pixels. Implementing more shader and texture units increase the performance of graphics card.
Render Output Pipelines (ROPs):
Render units, read the data from the memory and then blend the pixels then compress the data and write back to the memory and make the image ready to display.
Memory bus, bandwidth, clock and capacity:
Generally speaking, the memory on the graphics card is used for storing vertex, frame and textures buffers. Increasing the memory on the GPU allows you to play games at higher resolution, graphics quality and anti-aliasing settings, however, there always should be a balance between GPU's performance and the amount of memory. Assuming you have two identical low-end graphics card one with 1GB and the other with 2GB memory. If you play some games with both cards you may notice no actual performance gain on the 2GB graphics card because of the limitation of a low-end GPU to play games at higher resolution and quality settings.
Memory bus (interface) typically comes in three different form of 128-bit, 256-bit and 384-bit which is used to calculate the memory's bandwidth. The memory clock, is the frequency at which the memory works and memory bandwidth is calculated by the memory bus times by memory clock. Think of memory bus as width of a high way, a 128-bit means narrower and 384-bit the widest possible. A highway is designed with multiple lanes, each lanes can have certain width and speed limit. Memory clock, equals to lane speed limit and memory bus indicates total width of all lanes on the highway. A highway with more lanes gets more people to their places faster.
The GPU manufacturers usually compromise the speed for capacity. If you have come across two graphics cards having identical GPU, one with larger memory than the other and you have seen slightly better performance results from the card with smaller memory, don't be surprised! If you look closely at their specifications, you'll see that the card with larger memory comes with lower memory frequency than the other card and because the memory capacity can only be effective when extra texture rendering or resolution is required, therefore, the card with larger memory performs slower in games, so bigger isn't always better.
Now, let's look at the anti-aliasing (AA) and it's relation to memory. Anti-aliasing is a technique used to smooth the pixels or textures by blending the color of an edge with the color of pixels around it which results in more realistic look of the objects in games. Enabling MSAA and specially FSAA require a lot of GPU power and high memory capacity to be effective. FSAA forces the GPU to oversample or render the frame at resolution times the sampling rate). For example, playing a game with 4x FSAA at 1920x1080 resolution equals to playing the same game at 3840x2160 resolution with no AA enabled. Resolution also works the same way, increasing the resolution means more pixels need to be stored in the memory.
Some games incorporate specific features that works better with either Nvidia or AMD graphics card and that's mainly because of the methods used to optimize the game for GPU's architecture. A graphics card driver update is usually necessary to gain the best possible performance for new games.