I have 10.9.5 browser. Can I download the latest browser?

You need to upgrade to the latest version of macOS in order to accomplish what you want. That means your computer can be upgraded beyond where it is now.

How to upgrade to macOS Sierra

How to upgrade to macOS High Sierra

How to upgrade to macOS Mojave

Learn how to upgrade to the latest macOS

MacOS Mojave Compatibility List of Supported Macs

Apple has said that any Mac introduced in mid-2012 or later will support macOS Mojave, along with the 2010 and 2012 Mac Pro models if they have a Metal capable GPU. That’s fairly descriptive, but if you want a specific list of supported Mac hardware for macOS Mojave 10.14 then the following should be helpful to you:

• MacBook Pro (mid-2012 and newer)
• MacBook Air (mid-2012 and newer)
• MacBook (early 2015 and later)
• iMac (late 2012 or newer)
• iMac Pro (2017 or newer)
• Mac Pro (late 2013 or newer, or mid-2010 and mid-2012 models with Metal capable GPU)
• Mac Mini (late 2012 or newer)

As you can see from the list, basically every Mac that was introduced from mid-2012 onward supports macOS Mojave, along with a few earlier Mac Pro models. The supported hardware list for macOS 10.14 is a bit more restrictive than prior macOS system software releases, perhaps indicating that the latest software version will be more demanding on resources or simply just require particular system architecture components that are only available on more modern Macintosh hardware.