There are basically four different systems used to tighten cycling shoes: Velcro, ratchets, laces, and dials.
Almost all budget cycling shoes use Velcro straps for fastening, basically because it's a cheaper option. For the rider, Velcro straps do a good job of holding the shoe firmly in place but can be difficult to adjust on the move and it can also be a little tricky to get the precise fit you might want.
The other benefit of Velcro straps is that they are light. This means that they can also be found on very expensive lightweight shoes designed specifically for climbers. Over time, the Velcro can lose its effectiveness though, so that the hold on your foot can diminish.
The next step up in the fastening system food chain is ratchets, so look out for some of the great cycling shoes that offer these at a lower price point. A ratchet offers more precise adjustability than Velcro straps, gives a really secure fit, and can also easily be tightened while riding. However, loosening a ratchet is not quite as easily, often requiring two hands. Ratchets are generally found on mid-price shoes, and usually combined with a Velcro strap or two further down the shoe.
At the top of the tree you have dial closures, the fastening system used on most high-end shoes. In general, these are very secure, are easy to adjust, and can be dialled in (excuse the pun) to give a very precise hold. However, there are differences between different brands of dials. Boa dials offer the current gold standard with higher-end Boas providing incremental tightening and loosening, while those from other brands and lower spec Boas can't always be loosened quite so easily.
The other option on a few other high-end shoes such as the Giro Empires and Specialized S-Works Sub 6 shoes (as well as a few retro-style shoes) is laces. For those interested in marginal gains, laces are very aerodynamic but are also generally very comfy as there's wider distribution of pressure across the top of the foot and the laces self-adjust to help even it out. However, of course, laces are nigh-on impossible to adjust on the move.