When filing a trademark, the USPTO gives you two options: Standard Characters or Specialized Form (Stylized and/or Design). Let's look at each one in detail and that should help you decide the best way to file for your mark.
As with any discussion about filing for a trademark, it is always assumed that you've done your due diligence and had comprehensive research conducted. That being said...
This option is selected to register "word(s), letter(s), number(s), or any combination thereof, with no design element and when you are not claiming any particular font, style, size, or color, and absent any stylization or design element." In essence the USPTO is talking about plain text.
To qualify for this claim, the mark entered must fit within the standard character set. This includes letters and numbers but also some symbols, such as the ampersand (&), the dollar sign ($), the asterisk (*), etc.
Special Form (Stylized and/or Design)
This option is selected to register marks that are "comprised of stylized word(s), letter(s), and or number(s), and/or a design element." For example, if your business name is shown in a certain font and that font is an important element to your overall custom logo design, it might be a good idea to file it as such. This is also the selection you'll make if filing a business logo/design. A business logo can be filed on its own or with your name.
Can you File for Both?
Technically, yes though not at the same time. Two separate applications (meaning double the fees) would have to be filed if you wanted to choose the standard characters and the special form.
Why Would I File for Both?
Most small business owners would not but as with anything in the trademark world it is going to depend on your overall plans for the custom logo design. Let's say you have a name and a business logo - easy enough, you use the special form. But let's assume that your name is going to be shown without the business logo in a variety of fonts. Perhaps you have several product lines or your product line is geared towards different types of consumers and you have different font styles for each. Then it may be a good idea to also file the name as standard characters so as not to claim any one distinctive look to the mark but rather protect the name overall.