How to check if customer is logged in or not in Magento 2?

You first need to inject the following class in your constructor: /Magento/Customer/Model/Session :

protected $_customerSession;    // don't name this `$_session` since it is already used in \Magento\Customer\Model\Session and your override would cause problems

public function __construct(
    ...
    \Magento\Customer\Model\Session $session,
    ...
) {
    ...
    $this->_customerSession = $session;
    ...
}

Then in your class you can call the following:

if ($this->_customerSession->isLoggedIn()) {
    // Customer is logged in 
} else {
    // Customer is not logged in
}

In a template

It requires a bit more work in a template as you will have to setup a preference for the block that renders the template to do that the clean way:

<preference for="Block\That\Renders\The\Template"
            type="Vendor\Module\Block\Your\Custom\Block" />

Then in your custom block contrusctor you need to following the same dependency injection as for any class (explained above).

The extra step here is to create a public method that can be used in your template to check whether a customer is logged in or not

public function isCustomerLoggedIn()
{
    return $this->_customerSession->isLoggedIn();
}

Then in your template you can call:

if ($block->isCustomerLoggedIn()) {
    // Customer is logged in
} else {
    // Customer is not logged in
}

Alternative if the customer session is not initialized yet

There's another way of doing it which implies using Magento\Framework\App\Http\Context instead of Magento/Customer/Model/Session

Then you can call $this->_context->getValue(\Magento\Customer\Model\Context::CONTEXT_AUTH) instead of $this->_customerSession->isLoggedIn() to check whether the customer is logged in or not.