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The easiest way is to look at the selvedges and then remember same direction as selvedge is straight grain, perpendicular is cross grain 😎 But how do you determine the fabric grain if your fabric piece is missing selvage edges? You can test the stretch, remembering that fabric has more give on the cross grain 😉 So first look, with a magnifying glass if necessary, at the threads. Maybe even mark the two directions of the threads. Then stretch in the direction of the horizontal threads, and again in the direction of the vertical ones. Whichever way stretches or gives more is your crosswise grain. Once straight and cross grains are established, you can pull a thread along the straight grain as a reference for lining up your pattern pieces.Side note: I intentionally spelled selvedge two ways in this paragraph, as both are correct spellings, though selvage is the more commonly recognized one in the U.S. 
When you see grain arrows on your pattern pieces, they are usually intended to line up with the lengthwise grain on your fabric. (Sometimes it’s fine to use the cross grain instead, and I’ll talk more about that below.) It’s important to cut your garments so they are on grain because if they’re cut off grain they will torque, twist, and generally hang awkwardly. Grain is also important for a garment that’s cut on bias. A dress that’s cut on true bias will drape and cling nicely, but if it’s cut off grain, or at an angle that’s not 45 degrees from the length or cross grains, it will twist and look and feel unpleasant. (modified by Jesse Kelly from Bucheon, South Korea on December 3, 2020) 
There are three grains: straight grain, cross grain, and true bias. Straight grain, or lengthwise grain, are the threads going parallel to the selvedge of the fabric – the uncut edges that are bound so that they do not unravel. When fabric is cut at a shop, it had is cut on the crossgrain. The crossgrain are the threads running the width of the fabric – from one selvedge to the other. It is perpendicular to the straight grain. If you picture the straight grain being the longer threads that run in a straight line the entire length of the bolt of fabric and the crossgrain being the shorter threads that run across the width of the fabric, you’ll remember the difference. *Note – if you get a small yardage cut which is less than the width of your fabric (say, one yard if your fabric is 54″ wide), your lengthwise grain may actually be shorter than the cross grain! This does not matter. It is how the threads were woven that matters, not which is longer after a length of fabric is cut. (last edited 48 days ago by Maria Esquivel from Oslo, Norway) 
Based upon an article from thecuttingclass.com, the warp thread runs up and down while the weft thread runs right to left (You can rhyme weft with left to remember which one is which). The reason why these threads are important to the grainline is that they each react in different ways. For example, the warp thread is generally the stronger of the two and is the least likely to stretch out of shape. So for example on the straight front placket of a shirt, you don’t want it to go out of shape so if you align it with the strongest threads then it will hold it’s shape better. 
In knit fabric, the grain is there but it is different – it is called ‘direction’ in knits. Knit fabric consists of a series of loops. Knowing the grain or direction of the knit fabric is as important as in a woven fabric. In knit the direction refers to the placement of loops in the fabric. They are placed in lengthwise direction (courses) and crosswise direction(wales) with the greatest stretch being in the crosswise loop direction( mostly), with them placed perpendicular to each other just as for woven fabrics. A simple stretch test of the fabric will help you determine the direction of stretch on knits.