What does it mean when we say expenditure has been capitalised?

When expenditure is capitalised, the expenditure transaction has resulted in control over a noncurrent tangible asset and that instead of the transaction being recorded in full within the corresponding purchase period’s Income Statement, it will be recognised annually within Balance Sheet as a depreciated expense for the operating life of the asset. (Marshall et al 2010, Averkamp n.d.)
To be capitalised, the purchased asset must meet certain conditions: it will generate value for the business; is non-current as it will remain in the business for longer than 12 months; has a value that is significantly large to be of material interest (Marshall et al 2010). Most typically, expenditure will be capitalised for tangible assets such as land; buildings and equipment.
The capitalisation process occurs in accordance with the matching principle; depreciation costs are allocated within the periods of use and business benefit from the asset.

What does it mean to state that balance sheet values may not represent the current market values of long-lived assets?

Long-lived assets, specifically non-current tangible assets are commonly capitalised and are recorded as depreciated expense, as described above. (e.g. If a business purchases a truck costing $100,000 and it is expected to be used for 5 years, the business might have depreciation expense of $20,000 in each of the five years (Averkamp n.d.).)
Depreciation computation may differ between asset classes within a business and may also vary for tax purposes; be specified by law; or accounting standards (Wikipedia 2015) . Standard computation methods include: fixed percentage, straight line (as used in the example above), and declining balance (Marshall et al 2010).
Accumulated depreciation is used to calculate the assets book value (Averkamp n.d.). (e.g.